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A340757 Counterexamples to a conjecture of Ramanujan about congruences related to the partition function. 3
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 (list; graph; refs; listen; history; text; internal format)
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 √ľber Zerf√§llungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Eric Weisstein's World of Mathematics, Partition Function 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 A331613 A255626

Adjacent sequences:  A340754 A340755 A340756 * A340758 A340759 A340760

KEYWORD

nonn,easy

AUTHOR

Washington Bomfim, Jan 19 2021

STATUS

approved

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Last modified June 18 23:58 EDT 2021. Contains 345125 sequences. (Running on oeis4.)