A003598 Numbers of the form 5^i * 11^j.

1, 5, 11, 25, 55, 121, 125, 275, 605, 625, 1331, 1375, 3025, 3125, 6655, 6875, 14641, 15125, 15625, 33275, 34375, 73205, 75625, 78125, 161051, 166375, 171875, 366025, 378125, 390625, 805255, 831875, 859375, 1771561, 1830125, 1890625

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A108698, A003599, A107788, A108687, A108779, A108090.

Programs

GAP

Filtered([1..2*10^6],n->PowerMod(55,n,n)=0); # Muniru A Asiru, Mar 19 2019

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a003598 n = a003598_list !! (n-1)
a003598_list = f $ singleton (1,0,0) where
   f s = y : f (insert (5 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

Magma

[n: n in [1..2*10^6] | PrimeDivisors(n) subset [5, 11]]; // Vincenzo Librandi, Jun 27 2016

Mathematica

Take[Union[(5^#[[1]] 11^#[[2]])&/@Tuples[Range[0,20],{2}]],50]  (* Harvey P. Dale, Dec 26 2010 *)
fQ[n_]:=PowerMod[55, n, n] == 0; Select[Range[2*10^6], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

PARI

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(11),N=11^n;while(N<=lim,listput(v,N);N*=5));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

Sage

[n for n in (1..2*10^6) if n%55 in {0, 1, 5, 11, 15, 20, 25, 45} and all(x in {5,11} for x in prime_factors(n))] # F. Chapoton, Mar 16 2020

Formula

An asymptotic formula for a(n) is roughly 1/sqrt(55)*exp(sqrt(2*log(5)*log(11)*n)). - Benoit Cloitre, Mar 08 2002

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(55*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019

Sum_{n>=1} 1/a(n) = (5*11)/((5-1)*(11-1)) = 11/8. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(5)*log(11)*n)) / sqrt(55). - Vaclav Kotesovec, Sep 23 2020