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A036492 Offsets for the Atkin Partition Congruence theorem. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 29 23:37 EST 2020. Contains 338780 sequences. (Running on oeis4.)