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

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

Author

Olivier Gérard

Keywords

• nonn
• easy

References

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

Links

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

Crossrefs

Cf. A036491, A036492.

Programs

Haskell

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

Mathematica

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 *)

Formula

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

Extensions

Offset corrected by Reinhard Zumkeller, Feb 19 2013