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

Author

Olivier Gérard

Keywords

• nonn
• easy
• nice

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

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

Crossrefs

Cf. A000041, A036490, A036491.

Programs

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 *)

Formula

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

Extensions

Offset corrected by Reinhard Zumkeller, Feb 19 2013