A036491 -id:A036491 - cp's OEIS frontend

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

Olivier Gérard

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.

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.

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

Cf. A000041, A036490, A036491.

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 *)

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

Offset corrected by Reinhard Zumkeller, Feb 19 2013

A036490 Numbers whose prime factors are in {5, 7, 11}.

Original entry on oeis.org

5, 7, 11, 25, 35, 49, 55, 77, 121, 125, 175, 245, 275, 343, 385, 539, 605, 625, 847, 875, 1225, 1331, 1375, 1715, 1925, 2401, 2695, 3025, 3125, 3773, 4235, 4375, 5929, 6125, 6655, 6875, 8575, 9317, 9625, 12005, 13475, 14641, 15125, 15625, 16807

Offset: 1

Olivier Gérard

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 160.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A036491, A036492.

import Data.Set (Set, fromList, insert, deleteFindMin)
a036490 n = a036490_list !! (n-1)
a036490_list = f $ fromList [5,7,11] where
   f s = m : (f $ insert (5 * m) $ insert (7 * m) $ insert (11 * m) s')
         where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 19 2013

Select[Range[20000], (fi = FactorInteger[#][[All, 1]]; Intersection[fi, {5, 7, 11}] == fi)&]
(* or, for a large number of terms: *)
f[pp_(* primes *), max_(* maximum term *)] := Module[{a, aa, k, iter}, k = Length[pp]; aa = Array[a, k]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; A036490 = f[{5, 7, 11}, 2*10^14] // Rest (* Jean-François Alcover, Sep 19 2012, updated Nov 12 2016 *)

Sum_{n>=1} 1/a(n) = (5*7*11)/((5-1)*(7-1)*(11-1)) - 1 = 29/48. - Amiram Eldar, Sep 24 2020

a(n) ~ exp((6*log(5)*log(7)*log(11)*n)^(1/3)) / sqrt(385). - Vaclav Kotesovec, Sep 24 2020

Offset corrected by Reinhard Zumkeller, Feb 19 2013

cp's OEIS Frontend

A036492 Offsets for the Atkin Partition Congruence theorem.

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