# A SET OF SEQUENCES IN NUMBER THEORY

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1 A SET OF SEQUENCES IN NUMBER THEORY by Florentin Smarandache University of New Mexico Gallup, NM 87301, USA Abstract: New sequences are introduced in number theory, and for each one a general question: how many primes each sequence has. Keywords: sequence, symmetry, consecutive, prime, representation of numbers MSC: 11A67 Introduction. 74 new integer sequences are defined below, followed by references and some open questions. 1. Consecutive sequence: 1,12,123,1234,12345,123456, , , , , , , ,... How many primes are there among these numbers? In a general form, the Consecutive Sequence is considered in an arbitrary numeration base B. Reference: a) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache.

2 2. Circular sequence: 1,12,21,123,231,312,1234,2341,3412,4123,12345,23451,34512,45123,51234, ,234561,345612,456123,561234,612345, , , , How many primes are there among these numbers?

3 3. Symmetric sequence: 1,11,121,1221,12321,123321, , , , , , , , , , , , , , , , ,... How many primes are there among these numbers? In a general form, the Symmetric Sequence is considered in an arbitrary numeration base B. References: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA. b) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache.

4 4. Deconstructive sequence: 1,23,456,7891,23456,789123, , , , , How many primes are there among these numbers? Reference: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

5 5. Mirror sequence: 1,212,32123, , , , , , , , ,... Question: How many of them are primes?

6 6. Permutation sequence: 12,1342,135642, , , , , , , ,... Question: Is there any perfect power among these numbers? (Their last digit should be: either 2 for exponents of the form 4k+1, either 8 for exponents of the form 4k+3, where k >= 0.) Conjecture: no! 7. Generalizated permutation sequence: If g(n), as a function, gives the number of digits of a(n), and F if a permutation of g(n) elements, then: a(n) = F(1)F(2)...F(g(n)).

7 8. Simple numbers: 2,3,4,5,6,7,8,9,10,11,13,14,15,17,19,21,22,23,25,26,27,29,31, 33,34,35,37,38,39,41,43,45,46,47,49,51,53,55,57,58,61,62,65, 67,69,71,73,74,77,78,79,82,83,85,86,87,89,91,93,94,95,97,101,103,... (A number n is called <simple number> if the product of its proper divisors is less than or equal to n.) Generally speaking, n has the form: n = p, or p^2, or p^3, or pq, where p and q are distinct primes. How many primes, squares, and cubes are in this sequence? What interesting properties has it? Reference: a) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache.

8 9. Digital sum: 0,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,10,2,3,4,5,6,7,8,9,10,11, ,4,5,6,7,8,9,10,11,12,4,5,6,7,8,9,10,11,12,13,5,6,7,8,9,10,11,12,13,14, (d (n) is the sum of digits.) s How many primes, squares, and cubes are in this sequence? What interesting properties has it?

9 10. Digital products: 0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,2,4,6,8,19,12,14,16,18, ,3,6,9,12,15,18,21,24,27,0,4,8,12,16,20,24,28,32,36,0,5,10,15,20,25, (d (n) is the product of digits.) p How many primes, squares, and cubes are in this sequence? What interesting properties has it?

10 11. Code puzzle: ,202315, , , ,190924, , , ,200514, ,... Using the following letter-to-number code: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z then c (n) = the numerical code of the spelling of n in English p language; for exemple: 1 = ONE = , etc. Find a better codification (one sign only for each letter).

11 12. Pierced chain: 101, , , , , , ,... (c(n) = 101 * , for n >= 1) n-1 How many c(n)/101 are primes? References: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA. b) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache.

12 13. Divisor products: 1,2,3,8,5,36,7,64,27,100,11,1728,13,196,225,1024,17,5832,19, 8000,441,484,23,331776,125,676,729,21952,29,810000,31,32768, 1089,1156,1225, ,37,1444,1521, ,41,... (P (n) is the product of all positive divisors of n.) d How many primes, squares, and cubes are in this sequence? What interesting properties has it?

13 14. Proper divisor products: 1,1,1,2,1,6,1,8,3,10,1,144,1,14,15,64,1,324,1,400,21,22,1, 13824,5,26,27,784,1,27000,1,1024,33,34,35,279936,1,38,39, 64000,1,... (p (n) is the product of all positive divisors of n but n.) d How many primes, squares, and cubes are in this sequence? What interesting properties has it?

14 15. Square complements: 1,2,3,1,5,6,7,2,1,10,11,3,14,15,1,17,2,19,5,21,22,23,6,1,26, 3,7,29,30,31,2,33,34,35,1,37,38,39,10,41,42,43,11,5,46,47,3, 1,2,51,13,53,6,55,14,57,58,59,15,61,62,7,1,65,66,67,17,69,70,71,2,... Definition: for each integer n to find the smallest integer k such that nk is a perfect square.. (All these numbers are square free.) How many primes, squares, and cubes are in this sequence? What interesting properties has it?

15 16. Cubic complements: 1,4,9,2,25,36,49,1,3,100,121,18,169,196,225,4,289,12,361,50, 441,484,529,9,5,676,1,841,900,961,2,1089,1156,1225,6,1369, 1444,1521,25,1681,1764,1849,242,75,2116,2209,36,7,20,... Definition: for each integer n to find the smallest integer k such that nk is a perfect cub. (All these numbers are cube free.) How many primes, squares, and cubes are in this sequence? What interesting properties has it? 17. m-power complements: Definition: for each integer n to find the smallest integer k such that nk is a perfect m-power (m => 2). (All these numbers are m-power free.) How many primes, squares, and cubes are in this sequence? What interesting properties has it? Reference: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

16 18. Cube free sieve: 2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26, 28,29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,49,50, 51,52,53,55,57,58,59,60,61,62,63,65,66,67,68,69,70,71,73,... Definition: from the set of natural numbers (except 0 and 1): - take off all multiples of 2^3 (i.e. 8, 16, 24, 32, 40,...) - take off all multiples of 3^3 - take off all multiples of 5^3... and so on (take off all multiples of all cubic primes). (One obtains all cube free numbers.) How many primes, squares, and cubes are in this sequence? What interesting properties has it? 19. m-power free sieve: Definition: from the set of natural numbers (except 0 and 1) take off all multiples of 2^m, afterwards all multiples of 3^m,... and so on (take off all multiples of all m-power primes, m >= 2). (One obtains all m-power free numbers.) How many primes, squares, and cubes are in this sequence? What interesting properties has it?

17 20. Irrational root sieve: 2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,28, 29,30,31,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,50, 51,52,53,54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,... Definition: from the set of natural numbers (except 0 and 1): - take off all powers of 2^k, k >= 2, (i.e. 4, 8, 16, 32, 64,...) - take off all powers of 3^k, k >= 2; - take off all powers of 5^k, k >= 2; - take off all powers of 6^k, k >= 2; - take off all powers of 7^k, k >= 2; - take off all powers of 10^k, k >= 2;... and so on (take off all k-powers, k >= 2, of all square free numbers). One gets all square free numbers by the following method (sieve): from the set of natural numbers (except 0 and 1): - take off all multiples of 2^2 (i.e. 4, 8, 12, 16, 20,...) - take off all multiples of 3^2 - take off all multiples of 5^2... and so on (take off all multiples of all square primes); one obtains, therefore: 2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34, 35,37,38,39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66, 67,69,70,71,..., which are used for irrational root sieve. (One obtains all natural numbers those m-th roots, for any m >= 2, are irrational.) How many primes, squares, and cubes are in this sequence? What interesting properties has it?

18 References: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA. b) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache.

19 21. Odd sieve: 7,13,19,23,25,31,33,37,43,47,49,53,55,61,63,67,73,75,79,83, 85,91,93,97,... (All odd numbers that are not equal to the difference of two primes.) A sieve is used to get this sequence: - substract 2 from all prime numbers and obtain a temporary sequence; - choose all odd numbers that do not belong to the temporary one. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

20 22. Binary sieve: 1,3,5,9,11,13,17,21,25,27,29,33,35,37,43,49,51,53,57,59,65, 67,69,73,75,77,81,85,89,91,97,101,107,109,113,115,117,121, 123,129,131,133,137,139,145,149,... (Starting to count on the natural numbers set at any step from 1: - delete every 2-nd numbers - delete, from the remaining ones, every 4-th numbers... and so on: delete, from the remaining ones, every (2^k)-th numbers, k = 1, 2, 3,..) Conjectures: - there are an infinity of primes that belong to this sequence; - there are an infinity of numbers of this sequence which are not prime.

21 23. Trinary sieve: 1,2,4,5,7,8,10,11,14,16,17,19,20,22,23,25,28,29,31,32,34,35, 37,38,41,43,46,47,49,50,52,55,56,58,59,61,62,64,65,68,70,71, 73,74,76,77,79,82,83,85,86,88,91,92,95,97,98,100,101,103,104, 106,109,110,112,113,115,116,118,119,122,124,125,127,128,130, 131,133,137,139,142,143,145,146,149,... (Starting to count on the natural numbers set at any step from 1: - delete every 3-rd numbers - delete, from the remaining ones, every 9-th numbers... and so on: delete, from the remaining ones, every (3^k)-th numbers, k = 1, 2, 3,..) Conjectures: - there are an infinity of primes that belong to this sequence; - there are an infinity of numbers of this sequence which are not prime. 24. n-ary sieve (generalization, n >= 2): (Starting to count on the natural numbers set at any step from 1: - delete every n-th numbers; - delete, from the remaining ones, every (n^2)-th numbers;... and so on: delete, from the remaining ones, every (n^k)-th numbers, k = 1, 2, 3,..) Conjectures: - there are an infinity of primes that belong to this sequence; - there are an infinity of numbers of this sequence which

22 are not prime.

23 25. Consecutive sieve: 1,3,5,9,11,17,21,29,33,41,47,57,59,77,81,101,107,117,131, ,173,191,209,213,239,257,273,281,321,329,359,371,401,417, 441,435,491,... (From the natural numbers set: - keep the first number, delete one number out of 2 from all remaining numbers; - keep the first remaining number, delete one number out of 3 from the next remaining numbers; - keep the first remaining number, delete one number out of 4 from the next remaining numbers;... and so on, for step k (k >= 2): - keep the first remaining number, delete one number out of k from the next remaining numbers;..) This sequence is much less dense than the prime number sequence, and their ratio tends to p : n as n tends to infinity. n For this sequence we chosen to keep the first remaining number at all steps, but in a more general case: the kept number may be any among the remaining k-plet (even at random). How many primes, squares, and cubes are in this sequence? What interesting properties has it?

24 26. General-sequence sieve: Let u > 1, for i = 1, 2, 3,..., a stricly increasing i positive integer sequence. Then: From the natural numbers set: - keep one number among 1, 2, 3,..., u - 1, 1 and delete every u -th numbers; 1 - keep one number among the next u - 1 remaining numbers, 2 and delete every u -th numbers; 2... and so on, for step k (k >= 1): - keep one number among the next u - 1 remaining numbers, k and delete every u -th numbers; k.. Problem: study the relationship between sequence u, i = 1, 2, 3,..., i and the remaining sequence resulted from the general sieve. u, previously defined, is called sieve generator. i How many primes, squares, and cubes are in this sequence? What interesting properties has it?

25 27. Digital sequences: (This a particular case of sequences of sequences.) General definition: in any numeration base B, for any given infinite integer or rational sequence S, S, S,..., and any digit D from 0 to B-1, it's built up a new integer sequence witch associates to S the number of digits D of S in base B, 1 1 to S the number of digits D of S in base B, and so on For exemple, considering the prime number sequence in base 10, then the number of digits 1 (for exemple) of each prime number following their order is: 0,0,0,0,2,1,1,1,0,0,1,0,... (the digit-1 prime sequence). Second exemple if we consider the factorial sequence n! in base 10, then the number of digits 0 of each factorial number following their order is: 0,0,0,0,0,1,1,2,2,1,3,... (the digit-0 factorial sequence). Third exemple if we consider the sequence n^n in base 10, n=1,2,..., then the number of digits 5 of each term 1^1, 2^2, 3^3,..., following their order is: 0,0,0,1,1,1,1,0,0,0,... (The digit-5 n^n sequence) References: a) E. Grosswald, University of Pennsylvania, Philadelphia, Letter to the Author, August 3, 1985; b) R. K. Guy, University of Calgary, Alberta, Canada,

26 Letter to the Author, November 15, 1985; c) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

27 28. Construction sequences: (This a particular case of sequences of sequences.) General definition: in any numeration base B, for any given infinite integer or rational sequence S, S, S,..., and any digits D, D, , D (k < B), it's built up a new integer sequence such k that each of its terms Q < Q < Q <... is formed by these digits D, D,..., D only (all these digits are 1 2 k used), and matches a term S of the previous sequence. i For exemple, considering in base 10 the prime number sequence, and the digits 1 and 7 (for exemple), we construct a written-only-with-these-digits (all these digits are used) prime number new sequence: 17,71,... (the digit-1-7-only prime sequence). Second exemple, considering in base 10 the multiple of 3 sequence, and the digits 0 and 1, we construct a written-only-with-these-digits (all these digits are used) multiple of 3 new sequence: 1011,1101, 1110,10011,10101,10110,11001,11010,11100,... (the digit-0-1-only multiple of 3 sequence). How many primes, squares, and cubes are in this sequence? What interesting properties has it? References: a) E. Grosswald, University of Pennsylvania, Philadelphia,

28 Letter to F. Smarandache, August 3, 1985; b) R. K. Guy, University of Calgary, Alberta, Canada, Letter to F. Smarandache, November 15, 1985; c) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

29 29. General residual sequence: (x + C )...(x + C ), m = 2, 3, 4,..., 1 F(m) where C, 1 <= i <= F(m), forms a reduced set of i residues mod m, x is an integer, and F is Euler's totient. The General Residual Sequence is induced from the The Residual Function (see <Libertas Mathematica>): Let L : ZxZ --> Z be a function defined by L(x,m)=(x + C )...(x + C ), 1 F(m) where C, 1 <= i <= F(m), forms a reduced set of residues mod m, i m >= 2, x is an integer, and F is Euler's totient. The Residual Function is important because it generalizes the classical theorems by Wilson, Fermat, Euler, Wilson, Gauss, Lagrange, Leibnitz, Moser, and Sierpinski all together. For x=0 it's obtained the following sequence: L(m) = C... C, where m = 2, 3, 4,... 1 F(m) (the product of all residues of a reduced set mod m): 1,2,3,24,5,720,105,2240,189, ,385, ,19305, , , ,85085, , , , ,... which is found in "The Handbook of Integer Sequences", by N. J. A. Sloane, Academic Press, USA, The Residual Function extends it. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

30 References: a) F. Smarandache, "A numerical function in the congruence theory", in <Libertah Mathematica>, Texas State University, Arlington, 12, pp , 1992; see <Mathematical Reviews> 93i:11005 (11A07), p.4727, and <Zentralblatt fur Mathematik>, Band 773(1993/23), (11A); b) F. Smarandache, "Collected Papers" (Vol. 1), Ed. Tempus, Bucharest, 1994 (to appear); c) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA. d) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache.

31 30. (Inferior) prime part: 2,3,3,5,5,7,7,7,7,11,11,13,13,13,13,17,17,19,19,19,19,23,23, 23,23,23,23,29,29,31,31,31,31,31,31,37,37,37,37,41,41,43,43, 43,43,47,47,47,47,47,47,53,53,53,53,53,53,59,... (For any positive real number n one defines p (n) as the p largest prime number less than or equal to n.) 31. (Superior) prime part: 2,2,2,3,5,5,7,7,11,11,11,11,13,13,17,17,17,17,19,19,23,23, 23,23,29,29,29,29,29,29,31,31,37,37,37,37,37,37,41,41,41, 41,43,43,47,47,47,47,53,53,53,53,53,53,59,59,59,59,59,59,61,... (For any positive real number n one defines P (n) as the p smallest prime number greater than or equal to n.) Study these sequences. Reference: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

32 32. (Inferior) square part: 0,1,1,1,4,4,4,4,4,9,9,9,9,9,9,9,16,16,16,16,16,16,16,16,16, 25,25,25,25,25,25,25,25,25,25,25,36,36,36,36,36,36,36,36,36, 36,36,36,36,49,49,49,49,49,49,49,49,49,49,49,49,49,49,49,64,64,... (The largest square less than or equal to n.) 33. (Superior) square part: 0,1,4,4,4,9,9,9,9,9,16,16,16,16,16,16,16,25,25,25,25,25,25, 25,25,25,36,36,36,36,36,36,36,36,36,36,36,49,49,49,49,49,49, 49,49,49,49,49,49,49,64,64,64,64,64,64,64,64,64,64,64,64,64, 64,64,81,81,... (The smallest square greater than or equal to n.) Study these sequences.

33 34. (Inferior) cube part: 0,1,1,1,1,1,1,1,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,27,27, 27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27, 27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,64,64,64,... (The largest cube less than or equal to n.) 35. (Superior) cube part: 0,1,8,8,8,8,8,8,8,27,27,27,27,27,27,27,27,27,27,27,27,27, 27,27,27,27,27,27,64,64,64,64,64,64,64,64,64,64,64,64,64, 64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64, 64,64,64,64,64,125,125,125,.. (The smalest cube greater than or equal to n.) Study these sequences.

34 36. (Inferior) factorial part: 1,2,2,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,24,24,24,24, 24,24,24,24,24,24,24,24,24,24,24,24,24,24,... (F (n) is the largest factorial less than or equal to n.) p 37. (Superior) factorial part: 1,2,6,6,6,6,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24, 24,24,120,120,120,120,120,120,120,120,120,120,120,... (f (n) is the smallest factorial greater than or equal to n.) p Study these sequences.

35 38. Double factorial complements: 1,1,1,2,3,8,15,1,105,192,945,4,10395,46080,1,3, ,2560, ,192,5, , ,2,81081, , 35,23040, ,128, ,12,... (For each n to find the smallest k such that nk is a double factorial, i.e. nk = either 1*3*5*7*9*...*n if n is odd, either 2*4*6*8*...*n if n is even.) Study this sequence in interrelation with Smarandache function { S(n) is the smallest integer such that S(n)! is divisible by n ).

36 39. Prime additive complements: 1,0,0,1,0,1,0,3,2,1,0,1,0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0, 1,0,5,4,3,2,1,0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,... (For each n to find the smallest k such that n+k is prime.) Remark: is it possible to get as large as we want but finite decreasing k, k-1, k-2,..., 2, 1, 0 (odd k) sequence included in the previous sequence -- i.e. for any even integer are there two primes those difference is equal to it? I conjecture the answer is negative. Reference: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

37 40. Prime base: 0,1,10,100,101,1000,1001,10000,10001,10010,10100,100000, , , , , , , , , , , , , , , , ,... (Each number n written in the prime base.) (I define over the set of natural numbers the following infinite base: p = 1, and for k >= 1 p is the k-th prime number.) 0 k Every positive integer A may be uniquely written in the prime base as: n def --- A = (a... a a ) === \ a p, with all a = 0 or 1, (of course a = 1), n 1 0 (SP) / i i i n --- i=0 in the following way: - if p <= A < p then A = p + r ; n n+1 n 1 - if p <= r < p then r = p + r, m < n; m 1 m+1 1 m 2 and so on untill one obtains a rest r = 0. j Therefore, any number may be written as a sum of prime numbers + e, where e = 0 or 1. If we note by p(a) the superior part of A (i.e. the largest prime less than or equal to A), then A is written in the prime base as: A = p(a) + p(a-p(a)) + p(a-p(a)-p(a-p(a))) +.. This base is important for partitions with primes. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

38 41. Square base: 0,1,2,3,10,11,12,13,20,100,101,102,103,110,111,112,1000,1001, 1002,1003,1010,1011,1012,1013,1020,10000,10001,10002,10003, 10010,10011,10012,10013,10020,10100,10101,100000,100001,100002, ,100010,100011,100012,100013,100020,100100,100101,100102, ,100110,100111,100112,101000,101001,101002,101003,101010, ,101012,101013,101020,101100,101101,101102, ,... (Each number n written in the square base.) (I define over the set of natural numbers the following infinite base: for k >= 0 s = k^2.) k Every positive integer A may be uniquely written in the square base as: n def --- A = (a... a a ) === \ a s, with a = 0 or 1 for i >= 2, n 1 0 (S2) / i i i --- i=0 0 <= a <= 3, 0 <= a <= 2, and of course a = 1, 0 1 n in the following way: - if s <= A < s then A = s + r ; n n+1 n 1 - if s <= r < p then r = s + r, m < n; m 1 m+1 1 m 2 and so on untill one obtains a rest r = 0. j Therefore, any number may be written as a sum of squares (1 not counted as a square -- being obvious) + e, where e = 0, 1, or 3. If we note by s(a) the superior square part of A (i.e. the largest square less than or equal to A), then A is written in the square base as:

39 A = s(a) + s(a-s(a)) + s(a-s(a)-s(a-s(a))) +.. This base is important for partitions with squares. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

40 42. m-power base (generalization): (Each number n written in the m-power base, where m is an integer >= 2.) (I define over the set of natural numbers the following infinite m-power base: for k >= 0 t = k^m.) k Every positive integer A may be uniquely written in the m-power base as: n def --- A = (a... a a ) === \ a t, with a = 0 or 1 for i >= m, n 1 0 (SM) / i i i --- i= <= a <= ((i+2)^m - 1) / (i+1)^m (integer part) i for i = 0, 1,..., m-1, a = 0 or 1 for i >= m, and of course a = 1, i n in the following way: - if t <= A < t then A = t + r ; n n+1 n 1 - if t <= r < t then r = t + r, m < n; m 1 m+1 1 m 2 and so on untill one obtains a rest r = 0. j Therefore, any number may be written as a sum of m-powers (1 not counted as an m-power -- being obvious) + e, where e = 0, 1, 2,..., or 2^m-1. If we note by t(a) the superior m-power part of A (i.e. the largest m-power less than or equal to A), then A is written in the m-power base as: A = t(a) + t(a-t(a)) + t(a-t(a)-t(a-t(a))) +... This base is important for partitions with m-powers. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

41 43. Generalized base: (Each number n written in the generalized base.) (I define over the set of natural numbers the following infinite generalized base: 1 = g < g <... < g <..) 0 1 k Every positive integer A may be uniquely written in the generalized base as: n def A = (a... a a ) === \ a g, with 0 <= a <= (g - 1) / g n 1 0 (SG) / i i i -- i+1 i i=0 (integer part) for i = 0, 1,..., n, and of course a >= 1, n in the following way: - if g <= A < g then A = g + r ; n n+1 n 1 - if g <= r < g then r = g + r, m < n; m 1 m+1 1 m 2 and so on untill one obtains a rest r = 0. j If we note by g(a) the superior generalized part of A (i.e. the largest g less than or equal to A), then A is written in the i m-power base as: A = g(a) + g(a-g(a)) + g(a-g(a)-g(a-g(a))) +... This base is important for partitions: the generalized base may be any infinite integer set (primes, squares, cubes, any m-powers, Fibonacci/Lucas numbers, Bernoully numbers, etc.) those partitions are studied. A particular case is when the base verifies: 2g >= g i i+1 for any i, and g = 1, because all coefficients of a written 0 number in this base will be 0 or 1. i-1 Remark: another particular case: if one takes g = p, i = 1, 2, 3, i..., p an integer >= 2, one gets the representation of a number in the numerical base p {p may be 10 (decimal), 2 (binar), 16 (hexadecimal),

42 etc.}. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

43 44. Factorial quotients: 1,1,2,6,24,1,720,3,80,12, ,2, ,360,8,45, ,40, ,6,240, , ,1,145152, ,13440,180, ,... (For each n to find the smallest k such that nk is a factorial number.) Study this sequence in interrelation with Smarandache function. Reference: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

44 45. Double factorial numbers: 1,2,3,4,5,6,7,4,9,10,11,6,13,14,5,6,17,12,19,10,7,22,23,6, 15,26,9,14,29,10,31,8,11,34,7,12,37,38,13,10,41,14,43,22,9, 46,47,6,21,10,... (d (n) is the smallest integer such that d (n)!! is a f f multiple of n.) Study this sequence in interrelation with Smarandache function. Reference: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ , USA.

45 46. Primitive numbers (of power 2): 2,4,4,6,8,8,8,10,12,12,14,16,16,16,16,18,20,20,22,24,24,24, 26,28,28,30,32,32,32,32,32,34,36,36,38,40,40,40,42,44,44,46, 48,48,48,48,50,52,52,54,56,56,56,58,60,60,62,64,64,64,64,64,64,66,... (S (n) is the smallest integer such that S (n)! is divisible by 2^n.) 2 2 Curious property: this is the sequence of even numbers, each number being repeated as many times as its exponent (of power 2) is. This is one of irreductible functions, noted S (k), which 2 helps to calculate the Smarandache function. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

46 47. Primitive numbers (of power 3): 3,6,9,9,12,15,18,18,21,24,27,27,27,30,33,36,36,39,42,45,45, 48,51,54,54,54,57,60,63,63,66,69,72,72,75,78,81,81,81,81,84, 87,90,90,93,96,99,99,102,105,108,108,108,111,... (S (n) is the smallest integer such that S (n)! is divisible by 3^n.) 3 3 Curious property: this is the sequence of multiples of 3, each number being repeated as many times as its exponent (of power 3) is. This is one of irreductible functions, noted S (k), which helps 3 to calculate the Smarandache function. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

47 48. Primitive numbers (of power p, p prime) {generalization}: (S (n) is the smallest integer such that S (n)! is divisible by p^n.) p p Curious property: this is the sequence of multiples of p, each number being repeated as many times as its exponent (of power p) is. These are the irreductible functions, noted S (k), for any p prime number p, which helps to calculate the Smarandache function. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

48 49. Square residues: 1,2,3,2,5,6,7,2,3,10,11,6,13,14,15,2,17,6,19,10,21,22,23,6, 5,26,3,14,29,30,31,2,33,34,35,6,37,38,39,10,41,42,43,22,15, 46,47,6,7,10,51,26,53,6,14,57,58,59,30,61,62,21,... (s (n) is the largest square free number which divides n.) r Or, s (n) is the number n released of its squares: r if n = (p ^ a ) *... * (p ^ a ), with all p primes and all a >= 1, 1 1 r r i i then s (n) = p *... * p. r 1 r Remark: at least the (2^2)*k-th numbers (k = 1, 2, 3,...) are released of their squares; and more general: all (p^2)*k-th numbers (for all p prime, and k = 1, 2, 3,...) are released of their squares. How many primes, squares, and cubes are in this sequence? What interesting properties has it?

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