

SMARANDACHE PARTITION TYPE AND OTHER SEQUENCES*Eng. Dr. Fanel IACOBESCUElectrotechnic Faculty of Craiova, Romania
Examples of Smarandache Partition type sequences: A. 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, .... (How many times is n written as a sum of nonnull squares, disregarding the order of the terms: for example: 9 = 1^{2} + 1^{2} + 1^{2} + 1^{2} + 1^{2} + 1^{2} + 1^{2} + 1^{2} + 1^{2 } = 1^{2} + 1^{2} + 1^{2} + 1^{2} + 1^{2} +2^{2 } = 1^{2} + 2^{2} + 2^{2 } = 3^{2}, therefore ns(9) = 4.) B. 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, ... (How many times is n written as a sum of nonnull cubes, disregarding the order of the terms: for example: 9 = 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3 } = 1^{3} + 2^{3}, therefore, nc(9) = 2.) C. Generalpartition type sequence: Let f be an arithmetic function and R a relation among numbers. (How many times can n be written under the form: n = R(f(n_{1} ), f(n_{2} ), ..., f(n_{k} )) for some k and n_{1} , n_{2} , ..., n_{k} such that n_{1} + n_{2} + . . . + n_{k} = n? } Examples of other sequences: (1) Smarandache Antisymmetric sequence: 11, 1212, 123123, 12341234, 1234512345, 123456123456, 12345671234567, 1234567812345678, 123456789, 123456789, 1234567891012345678910, 1234567891011, 1234567891011, ... (2) Smarandache Triangular base: 1, 2, 10, 11, 12, 100, 101, 102, 110, 1000, 1001, 1002, 1010, 1011, 10000, 10001, 10002, 10010, 10011, 10012, 100000, 100001, 100002, 100010, 100011, 100012, 100100, 1000000, 1000001, 1000002, 1000010, 1000011, 1000012, 1000100, ... (Numbers written in the triangular base, defined as follows: t(n) = (n(n+1))/2, for n >= 1.) (3) Smarandache Double factorial base: 1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, 1100, 1101, 1110, 1200, 10000, 10001, 10010, 10100, 10101, 10110, 10200, 10201, 11000, 11001, 11010, 11100, 11101, 11110, 11200, 11201, 12000, ... (Numbers written in the double factorial base, defined as follows: df(n) = n!!) (4) Smarandache Nonmultiplicative sequence: General definition: Let m_{1}, m_{2}, ..., m_{k} be the first k terms of the sequence, where k >= 2; then m_{i} , for i >= k+1, is the smallest number not equal to the product of k previous distinct terms. (5) Smarandache Nonarithmetic progression: 1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41, 64, ... General definition: if m_{1} , m_{2}, are the first two terms of the sequence, then m_{k} , for k >= 3, is the smallest number such that no 3term arithmetic progression is in the sequence. In our case the first two terms are 1, respectively 2. Generalization: same initial conditions, but no iterm arithmetic progression in the sequence (for a given i >= 3). (6) Smarandache Prime product sequence: 2, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, 7420738134811, 304250263527211, ... P_{n} = 1 + p_{1} p_{2} . . . p_{n} , where p_{k} is the kth prime. Question: How many of them are prime? (7) Smarandache Square product sequence: 2, 5, 37, 577, 14401, 518401, 25401601, 1625702401, 131681894401, 13168189440001, 1593350922240001, ... S_{n} = 1 + s_{1} s_{2} . . . s_{n} , where s_{k} is the kth square number. Question: How many of them are prime? (8) Smarandache Cubic product sequence: 2, 9, 217, 13825, 1728001, 373248001, 128024064001, 65548320768001, ... C_{k} = 1 + c_{1} c_{2} ...c_{n} , where c_{k} is the kth cubic number. Question: How many of them are prime? (9) Smarandache Factorial product sequence: 2, 3, 13, 289, 34561, 24883201, 125411328001, 5056584744960001, ... F_{n} = 1 + f_{1} f_{2} ...f_{n} , where f_{k} is the kth factorial number. Question: How many of them are prime? (10) Smarandache Uproduct sequence {generalization}: Let u_{n} , n >= 1, be a positive integer sequence. Then we define a Usequence as follows: U_{n} = 1 + u_{1} u_{2} . . . u_{n} . (11) Smarandache Nongeometric progression. 1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48, 50, 51, 53, . . . General definition: if m_{1} ,m_{2}, are the first two terms of the sequence, then m_{k}, for k >= 3, is the smallest number such that no 3term geometric progression is in the sequence. In our case the first two terms are 1, respectively 2. (12) Smarandache Unary sequence: 11, 111, 11111, 1111111, 11111111111, 1111111111111, 1111111111111111, 1111111111111111111, 11111111111111111111111, 11111111111111111111111111111, 1111111111111111111111111111111, ... u(n) = 11...1, p_{n} digits of "1", where p_{n} is the nth prime. The old question: are there are infinite number of primes belonging to the sequence? (13) Smarandache Noprimedigit sequence: 1, 4, 6, 8, 9, 10. 11, 1, 1, 14, 1, 16. 1, 18, 19, 0, 1, 4, 6, 8, 9, 0, 1, 4, 6, 8, 9, 40, 41, 42, 4, 44, 4, 46, 48, 49, 0, ... (Take out all prime digits of n.) (14) Smarandache Nosquaredigitsequence. 2, 3, 5, 6, 7, 8, 2, 3, 5, 6, 7, 8, 2, 2, 22, 23, 2, 25, 26, 27, 28, 2, 3, 3, 32, 33, 3, 35, 36, 37, 38, 3, 2, 3, 5, 6, 7, 8, 5, 5, 52, 53, 5, 55, 56, 57, 58, 5, 6, 6, 62, ... (Take out all square digits of n.) * This paper first appeared in Bulletin of Pure and Applied Sciences, Vol. 16 E(No. 2) 1997; P. 237240. 