# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a340757 Showing 1-1 of 1 %I A340757 #121 Feb 24 2023 02:36:39 %S A340757 243,586,1272,2301,2644,2987,3673,4702,5045,5388,6074,7103,7446,7789, %T A340757 8475,9504,9847,10190,10876,11905,12248,12591,13277,14306,14649,14992, %U A340757 15678,16707,17050,17393,18079,19108,19451,19794,20480,21509,21852,22195,22881,23910 %N A340757 Counterexamples to a conjecture of Ramanujan about congruences related to the partition function. %C A340757 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. %C A340757 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). %C A340757 See A340957 for the converse, those n == 243 (mod 7^3) where the conjecture does hold. %H A340757 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) %H A340757 A. O. L. Atkin and S. M. Hussain, Some Properties of Partitions II, Trans. Amer. Math. Soc. 89, pp. 184-200 (1958). %H A340757 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. %H A340757 D. H. Lehmer, On a conjecture of Ramanujan, J. London Math. Soc. 11, 114-118 (1936). %H A340757 D. H. Lehmer, On the Hardy-Ramanujan Series for the partition function,, J. London Math. Soc. 12, 171-176 (1937). %H A340757 G. N. Watson, A New Proof of the Rogers-Ramanujan Identities, J. London Math. Soc., V. s1-4, 1, Pages 4-9. (1929). %H A340757 G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. %H A340757 Eric Weisstein's World of Mathematics, Partition Function P Congruences. %H A340757 Index entries for related partition-counting sequences %e A340757 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). %o A340757 (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++)) }; %o A340757 seq(100); \\ Gives the first 100 terms of the sequence. %Y A340757 Cf. A000041, A052462, A052465, A052466, A182719, A340957. %K A340757 nonn,easy %O A340757 1,1 %A A340757 _Washington Bomfim_, Jan 19 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE