login
Numbers whose prime factors are raised to the fifth power.
10

%I #31 Sep 14 2024 10:49:23

%S 32,243,3125,7776,16807,100000,161051,371293,537824,759375,1419857,

%T 2476099,4084101,5153632,6436343,11881376,20511149,24300000,28629151,

%U 39135393,45435424,52521875,69343957,79235168,90224199,115856201

%N Numbers whose prime factors are raised to the fifth power.

%F Sum_{k>=1} 1/a(k) = zeta(5)/zeta(10) - 1 = A157291 - 1. - _Amiram Eldar_, May 22 2020

%F a(n) = A005117(n+1)^5. - _Chai Wah Wu_, Sep 13 2024

%e 7776 = 32*243 = 2^5*3^5 so the prime factors, 2 and 3, are raised to the fifth power.

%t Select[ Range@41^5, Union[Last /@ FactorInteger@# ] == {5} &] (* _Robert G. Wilson v_ *)

%t Rest[Select[Range[100], SquareFreeQ]^5] (* _Vaclav Kotesovec_, May 22 2020 *)

%o (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 (Python)

%o from math import isqrt

%o from sympy import mobius

%o def A113850(n):

%o def f(x): return int(n+x+1-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))

%o m, k = n, f(n)

%o while m != k: m, k = k, f(k)

%o return m**5 # _Chai Wah Wu_, Sep 13 2024

%Y Proper subset of A000584.

%Y Nonunit terms of A329332 column 5 in ascending order.

%Y Cf. A005117, A157291.

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Jan 25 2006

%E More terms from _Robert G. Wilson v_, Jan 26 2006

%E Offset corrected by _Chai Wah Wu_, Sep 13 2024