A340757 Counterexamples to a conjecture of Ramanujan about congruences related to the partition function.

243, 586, 1272, 2301, 2644, 2987, 3673, 4702, 5045, 5388, 6074, 7103, 7446, 7789, 8475, 9504, 9847, 10190, 10876, 11905, 12248, 12591, 13277, 14306, 14649, 14992, 15678, 16707, 17050, 17393, 18079, 19108, 19451, 19794, 20480, 21509, 21852, 22195, 22881, 23910

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

Table of n, a(n) for n=1..40.

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.

Index entries for related partition-counting sequences

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

Cf. A000041, A052462, A052465, A052466, A182719, A340957.

Sequence in context: A157958 A232924 A067838 * A255111 A353320 A331613

Adjacent sequences: A340754 A340755 A340756 * A340758 A340759 A340760

Keyword

nonn,easy

Author

Washington Bomfim, Jan 19 2021

Status

approved