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%I #23 Nov 05 2020 09:40:35
%S 4,5,6,24,19,47,39,61,116,99,124,194,149,243,369,292,479,599,600,474,
%T 1174,721,974,929,1524,2301,1909,2899,2474,2987,2294,3099,5682,4849,
%U 4714,3724,6074,7376,9224,9504,7299,14031,11974,14974,11905,18079,14999,11849,14306,23469,29349,18024,24349
%N Offsets for the Atkin Partition Congruence theorem.
%C The Atkin Theorem, inspired by a famous conjecture of Ramanujan, gives congruences properties of certain partition numbers, generalizing many previous results.
%C Let T = 5^a*7^b*11^c (A036490) and Q = 5^a*7^(floor[(b+2)/2])*11^c (A036491).
%C If 24*g = 1 mod T, then p(g) = 0 mod Q, where p(g) is the number of integer partitions of g.
%C In fact, for k >= 0, p(g + k*T) = 0 mod Q, since 24*(g + k*T) = 24*g = 1 mod T.
%C A036492(n) lists the smallest g for basis T = A036490(n) and modulus Q = A036491(n).
%C 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.
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, pp. 159-161.
%D 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.
%H A. O. L. Atkin, <a href="https://doi.org/10.1017/S0017089500000045">Proof of a conjecture of Ramanujan</a>, Glasgow Math. J., 8 (1967), 14-32.
%F 24 * a(n) == 1 (mod A036490(n)). - _Sean A. Irvine_, Nov 04 2020
%t 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 *)
%Y Cf. A000041, A036490, A036491.
%K nonn,easy,nice
%O 1,1
%A _Olivier Gérard_
%E Offset corrected by _Reinhard Zumkeller_, Feb 19 2013
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