cp's OEIS Frontend

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

%I A003595 #56 Jul 06 2025 10:34:17
%S A003595 1,5,7,25,35,49,125,175,245,343,625,875,1225,1715,2401,3125,4375,6125,
%T A003595 8575,12005,15625,16807,21875,30625,42875,60025,78125,84035,109375,
%U A003595 117649,153125,214375,300125,390625,420175,546875,588245,765625,823543,1071875,1500625
%N A003595 Numbers of the form 5^i*7^j with i, j >= 0.
%C A003595 Successive k such that phi(35*k) = 24*k: 35*a(n) = A033851(n). - _Artur Jasinski_, Nov 09 2008
%H A003595 Vaclav Kotesovec, <a href="/A003595/b003595.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%H A003595 Vaclav Kotesovec, <a href="/A003595/a003595.jpg">Graph - the asymptotic ratio (400000 terms)</a>.
%H A003595 David Ryan, <a href="https://arxiv.org/abs/1612.01860">An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation</a>, Preprint, arXiv:1612.01860 [cs.SD], 2016-2017.
%F A003595 Sum_{n>=1} 1/a(n) = (5*7)/((5-1)*(7-1)) = 35/24. - _Amiram Eldar_, Sep 22 2020
%F A003595 a(n) ~ exp(sqrt(2*log(5)*log(7)*n)) / sqrt(35). - _Vaclav Kotesovec_, Sep 22 2020
%F A003595 a(n) = 5^A025652(n) * 7^A025667(n). - _R. J. Mathar_, Jul 06 2025
%t A003595 a = {}; Do[If[EulerPhi[35 k] == 24 k, AppendTo[a, k]], {k, 1, 10000}]; a (* _Artur Jasinski_, Nov 09 2008 *)
%t A003595 fQ[n_] := PowerMod[35, n, n] == 0; Select[Range[600000], fQ] (* _Bruno Berselli_, Sep 24 2012 *)
%o A003595 (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
%o A003595 (Magma) [n: n in [1..600000] | PrimeDivisors(n) subset [5,7]]; // _Bruno Berselli_, Sep 24 2012
%o A003595 (Haskell)
%o A003595 import Data.Set (singleton, deleteFindMin, insert)
%o A003595 a003595 n = a003595_list !! (n-1)
%o A003595 a003595_list = f $ singleton 1 where
%o A003595    f s = y : f (insert (5 * y) $ insert (7 * y) s')
%o A003595                where (y, s') = deleteFindMin s
%o A003595 -- _Reinhard Zumkeller_, May 16 2015
%o A003595 (Python)
%o A003595 from sympy import integer_log
%o A003595 def A003595(n):
%o A003595     def bisection(f,kmin=0,kmax=1):
%o A003595         while f(kmax) > kmax: kmax <<= 1
%o A003595         while kmax-kmin > 1:
%o A003595             kmid = kmax+kmin>>1
%o A003595             if f(kmid) <= kmid:
%o A003595                 kmax = kmid
%o A003595             else:
%o A003595                 kmin = kmid
%o A003595         return kmax
%o A003595     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))
%o A003595     return bisection(f,n,n) # _Chai Wah Wu_, Sep 16 2024
%Y A003595 Cf. A033851, A143207, A147571-A147580.
%Y A003595 Cf. A003586, A003592, A003593, A003594.
%Y A003595 Cf. A025652, A025667.
%K A003595 nonn
%O A003595 1,2
%A A003595 _N. J. A. Sloane_