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A113850 Numbers whose prime factors are raised to the fifth power.

%I A113850 #31 Sep 14 2024 10:49:23
%S A113850 32,243,3125,7776,16807,100000,161051,371293,537824,759375,1419857,
%T A113850 2476099,4084101,5153632,6436343,11881376,20511149,24300000,28629151,
%U A113850 39135393,45435424,52521875,69343957,79235168,90224199,115856201
%N A113850 Numbers whose prime factors are raised to the fifth power.
%F A113850 Sum_{k>=1} 1/a(k) = zeta(5)/zeta(10) - 1 = A157291 - 1. - _Amiram Eldar_, May 22 2020
%F A113850 a(n) = A005117(n+1)^5. - _Chai Wah Wu_, Sep 13 2024
%e A113850 7776 = 32*243 = 2^5*3^5 so the prime factors, 2 and 3, are raised to the fifth power.
%t A113850 Select[ Range@41^5, Union[Last /@ FactorInteger@# ] == {5} &] (* _Robert G. Wilson v_ *)
%t A113850 Rest[Select[Range[100], SquareFreeQ]^5] (* _Vaclav Kotesovec_, May 22 2020 *)
%o A113850 (PARI) allpwrfact(n,p) = \All prime factors are raised to the power p { local(x,j,ln,y,flag); for(x=4,n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1,ln, if(y[2][j]==p,flag++); ); if(flag==ln,print1(x",")); ) }
%o A113850 (Python)
%o A113850 from math import isqrt
%o A113850 from sympy import mobius
%o A113850 def A113850(n):
%o A113850     def f(x): return int(n+x+1-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A113850     m, k = n, f(n)
%o A113850     while m != k: m, k = k, f(k)
%o A113850     return m**5 # _Chai Wah Wu_, Sep 13 2024
%Y A113850 Proper subset of A000584.
%Y A113850 Nonunit terms of A329332 column 5 in ascending order.
%Y A113850 Cf. A005117, A157291.
%K A113850 easy,nonn
%O A113850 1,1
%A A113850 _Cino Hilliard_, Jan 25 2006
%E A113850 More terms from _Robert G. Wilson v_, Jan 26 2006
%E A113850 Offset corrected by _Chai Wah Wu_, Sep 13 2024