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
Keywords
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
Programs
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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
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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
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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
Comments