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%I #121 Feb 24 2023 02:36:39
%S 243,586,1272,2301,2644,2987,3673,4702,5045,5388,6074,7103,7446,7789,
%T 8475,9504,9847,10190,10876,11905,12248,12591,13277,14306,14649,14992,
%U 15678,16707,17050,17393,18079,19108,19451,19794,20480,21509,21852,22195,22881,23910
%N Counterexamples to a conjecture of Ramanujan about congruences related to the partition function.
%C 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 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 See A340957 for the converse, those n == 243 (mod 7^3) where the conjecture does hold.
%H A. O. L. Atkin and P. Swinnerton-Dyer, <a href="https://doi.org/10.1112/plms/s3-4.1.84">Some Properties of Partitions,</a>, Proceedings of the London Math. Soc., V. s3-4, Issue 1, pp. 84-106, (1954)
%H A. O. L. Atkin and S. M. Hussain, <a href="https://doi.org/10.1090/S0002-9947-1958-0103872-3">Some Properties of Partitions II</a>, Trans. Amer. Math. Soc. 89, pp. 184-200 (1958).
%H Hansraj Gupta, <a href="https://nvlpubs.nist.gov/nistpubs/jres/74B/jresv74Bn1p1_A1b.pdf">Partitions - A Survey</a>, Journal of Research of the National Bureau of Standards - B. Mathematical Sciences Vol. 74B, No. 1, January-March 1970. See section 6.1.
%H D. H. Lehmer, <a href="https://doi.org/10.1112/jlms/s1-11.2.114">On a conjecture of Ramanujan</a>, J. London Math. Soc. 11, 114-118 (1936).
%H D. H. Lehmer, <a href="https://doi.org/10.1112/jlms/s1-12.2.171">On the Hardy-Ramanujan Series for the partition function,</a>, J. London Math. Soc. 12, 171-176 (1937).
%H G. N. Watson, <a href="https://doi.org/10.1112/jlms/s1-4.1.4">A New Proof of the Rogers-Ramanujan Identities</a>, J. London Math. Soc., V. s1-4, 1, Pages 4-9. (1929).
%H G. N. Watson, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002174499">Ramanujans Vermutung über Zerfällungsanzahlen</a>. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionPCongruences.html">Partition Function P Congruences</a>.
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%e 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 (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 seq(100); \\ Gives the first 100 terms of the sequence.
%Y Cf. A000041, A052462, A052465, A052466, A182719, A340957.
%K nonn,easy
%O 1,1
%A _Washington Bomfim_, Jan 19 2021
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