cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A036491 Transformation of A036490: 5^a*7^b*11^c -> 5^a*7^floor((b+2)/2)*11^c.

Original entry on oeis.org

5, 7, 11, 25, 35, 49, 55, 77, 121, 125, 175, 245, 275, 49, 385, 539, 605, 625, 847, 875, 1225, 1331, 1375, 245, 1925, 343, 2695, 3025, 3125, 539, 4235, 4375, 5929, 6125, 6655, 6875, 1225, 9317, 9625, 1715, 13475, 14641, 15125, 15625, 343, 2695, 21175
Offset: 1

Views

Author

Olivier Gérard

Keywords

References

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

Links

Crossrefs

Cf. A036490, A036492.

Programs

  • Haskell
    a036491 n = f z z where
   f x y | x `mod` 2401 == 0 = f (x `div` 49) (y `div` 7)
         | x `mod` 343 == 0  = y `div` 7
         | otherwise         = y
   z = a036490 n
-- Reinhard Zumkeller, Feb 19 2013
  • Mathematica
    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; a[n_] := (a0 = A036490[[n]]; b = Max[1, IntegerExponent[a0, 7]]; 7^(Floor[(b+2)/2]-b) * a0); Table[a[n], {n, 1, Length[A036490]}]; (* Jean-François Alcover, Sep 19 2012, updated Nov 12 2016 *)

Extensions

Offset corrected by Reinhard Zumkeller, Feb 19 2013

A036492 Offsets for the Atkin Partition Congruence theorem.

Original entry on oeis.org

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

Views

Author

Olivier Gérard

Keywords

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

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
Showing 1-2 of 2 results.

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