A003595 Numbers of the form 5^i*7^j with i, j >= 0.

1, 5, 7, 25, 35, 49, 125, 175, 245, 343, 625, 875, 1225, 1715, 2401, 3125, 4375, 6125, 8575, 12005, 15625, 16807, 21875, 30625, 42875, 60025, 78125, 84035, 109375, 117649, 153125, 214375, 300125, 390625, 420175, 546875, 588245, 765625, 823543, 1071875, 1500625

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Successive k such that phi(35*k) = 24*k: 35*a(n) = A033851(n). - Artur Jasinski, Nov 09 2008

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
• Vaclav Kotesovec, Graph - the asymptotic ratio (400000 terms).
• David Ryan, An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation, Preprint, arXiv:1612.01860 [cs.SD], 2016-2017.

Crossrefs

Cf. A033851, A143207, A147571-A147580.

Cf. A003586, A003592, A003593, A003594.

Cf. A025652, A025667.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a003595 n = a003595_list !! (n-1)
a003595_list = f $ singleton 1 where
   f s = y : f (insert (5 * y) $ insert (7 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

Magma

[n: n in [1..600000] | PrimeDivisors(n) subset [5,7]]; // Bruno Berselli, Sep 24 2012

Mathematica

a = {}; Do[If[EulerPhi[35 k] == 24 k, AppendTo[a, k]], {k, 1, 10000}]; a (* Artur Jasinski, Nov 09 2008 *)
fQ[n_] := PowerMod[35, n, n] == 0; Select[Range[600000], fQ] (* Bruno Berselli, Sep 24 2012 *)

PARI

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N*=5));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

Python

from sympy import integer_log
def A003595(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//7**i,5)[0]+1 for i in range(integer_log(x,7)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Formula

Sum_{n>=1} 1/a(n) = (5*7)/((5-1)*(7-1)) = 35/24. - Amiram Eldar, Sep 22 2020

a(n) ~ exp(sqrt(2*log(5)*log(7)*n)) / sqrt(35). - Vaclav Kotesovec, Sep 22 2020

a(n) = 5^A025652(n) * 7^A025667(n). - R. J. Mathar, Jul 06 2025