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%I #55 Jul 02 2022 09:28:37
%S 25,100,125,225,400,625,900,1000,1225,1600,2025,2500,3025,3125,3375,
%T 3600,4225,4900,5625,6400,7225,8000,8100,9025,10000,11025,12100,13225,
%U 14400,15625,16900,18225,19600,21025,22500,24025,25600,27000,27225,28900,30625
%N Perfect powers that are divisible by 5.
%C a(n) == 0 (mod 25).
%C a(n) == {0, 25, 75} (mod 100).
%H Michael De Vlieger, <a href="/A353152/b353152.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = (5*m)^k for some positive m and some k > 1. - _David A. Corneth_, Apr 28 2022
%F Sum_{n>=1} 1/a(n) = -Sum_{k>=2} mu(k)*zeta(k)/5^k = 0.0756933503... - _Amiram Eldar_, Jul 02 2022
%e 225 is a term since 225 = 15^2 is a power of a multiple of 5.
%t Select[Range[32000], And[GCD @@ FactorInteger[#][[All, 2]] > 1, Divisible[#, 5]] &] (* _Michael De Vlieger_, Apr 30 2022 *)
%o (PARI) isok(k) = ispower(k) && !(k%5); \\ _Michel Marcus_, Apr 27 2022
%o (PARI) upto(n) = { my(res = List()); forstep(i = 5, sqrtint(n), 5, c = i; for(e = 2, logint(n, i), c*=i; listput(res, c) ) ); Set(res) } \\ _David A. Corneth_, Apr 27 2022
%Y Cf. A000351, A011557.
%Y Intersection of A001597 and A008587.
%Y Cf. A075090 (even perfect powers).
%K nonn,easy
%O 1,1
%A _Marco Ripà_, Apr 26 2022
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