A036492 Offsets for the Atkin Partition Congruence theorem.

4, 5, 6, 24, 19, 47, 39, 61, 116, 99, 124, 194, 149, 243, 369, 292, 479, 599, 600, 474, 1174, 721, 974, 929, 1524, 2301, 1909, 2899, 2474, 2987, 2294, 3099, 5682, 4849, 4714, 3724, 6074, 7376, 9224, 9504, 7299, 14031, 11974, 14974, 11905, 18079, 14999, 11849, 14306, 23469, 29349, 18024, 24349

Offset

1,1

Comments

The Atkin Theorem, inspired by a famous conjecture of Ramanujan, gives congruences properties of certain partition numbers, generalizing many previous results.

Let T = 5^a*7^b*11^c (A036490) and Q = 5^a*7^(floor[(b+2)/2])*11^c (A036491).

If 24*g = 1 mod T, then p(g) = 0 mod Q, where p(g) is the number of integer partitions of g.

In fact, for k >= 0, p(g + k*T) = 0 mod Q, since 24*(g + k*T) = 24*g = 1 mod T.

A036492(n) lists the smallest g for basis T = A036490(n) and modulus Q = A036491(n).

The first case using the full force of the theorem is for n = 46 corresponding to T = 5*7^3*11 = 18865, giving Q = 2695 and g = 18079.

References

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, pp. 159-161.

G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, Collected Papers of S. Ramanujan, CUP, 1927, #25 (1919), pp. 210-213, and #28 (1919), p. 230.

Links

Table of n, a(n) for n=1..53.

A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32.

Formula

24 * a(n) == 1 (mod A036490(n)). - Sean A. Irvine, Nov 04 2020

Mathematica

Map[Function[df, First@Select[Range[3, df], Mod[24 #, df] == 1 &, 1]],  Select[Range[40000], DeleteCases[FactorInteger[#], {5|7|11, _}] == {} &]] (* From Olivier Gérard, Nov 12 2016 *)

Crossrefs

Cf. A000041, A036490, A036491.

Sequence in context: A109889 A103518 A103313 * A048075 A048016 A287648

Adjacent sequences: A036489 A036490 A036491 * A036493 A036494 A036495

Keyword

nonn,easy,nice

Author

Olivier Gérard

Extensions

Offset corrected by Reinhard Zumkeller, Feb 19 2013

Status

approved