OFFSET
1,1
COMMENTS
For b in 5,7,11, and all integers n,e >= 1, Ramanujan conjectured that if (24*n-1) is divisible by b^e, the partition function p(n) = A000041(n) is also divisible by b^e.
Chowla found the first counterexample a(1) = 243. Watson showed the conjecture holds for b=5, and Atkin showed it holds for b=11. Watson showed p(n) is divisible by 7^floor((d+2)/2) when 24n-1 is divisible by 7^d, so that exceptions here are restricted to 24n-1 == 0 (mod 7^3), which is n == 243 (mod 7^3).
See A340957 for the converse, those n == 243 (mod 7^3) where the conjecture does hold.
LINKS
A. O. L. Atkin and P. Swinnerton-Dyer, Some Properties of Partitions,, Proceedings of the London Math. Soc., V. s3-4, Issue 1, pp. 84-106, (1954)
A. O. L. Atkin and S. M. Hussain, Some Properties of Partitions II, Trans. Amer. Math. Soc. 89, pp. 184-200 (1958).
Hansraj Gupta, Partitions - A Survey, Journal of Research of the National Bureau of Standards - B. Mathematical Sciences Vol. 74B, No. 1, January-March 1970. See section 6.1.
D. H. Lehmer, On a conjecture of Ramanujan, J. London Math. Soc. 11, 114-118 (1936).
D. H. Lehmer, On the Hardy-Ramanujan Series for the partition function,, J. London Math. Soc. 12, 171-176 (1937).
G. N. Watson, A New Proof of the Rogers-Ramanujan Identities, J. London Math. Soc., V. s1-4, 1, Pages 4-9. (1929).
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
EXAMPLE
243 is a term because for n = 243, the condition of Ramanujan (24*n - 1) divisible by b^e is true, and p(n) is not divisible by (b^e). [We have base b=7, and exponent e=3 in this case.] Since a(1) = A182719(91), 90 numbers satisfy the conjecture before the first counterexample a(1).
PROG
(PARI) seq(x) = {my( n = -100, N=0); while(N < x, n += 343; if(valuation(numbpart(n), 7) < valuation(24*n-1, 7), print1(n", "); N++)) };
seq(100); \\ Gives the first 100 terms of the sequence.
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Washington Bomfim, Jan 19 2021
STATUS
approved