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If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = sum_{p|n} b(p,n)^b(p,n).
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%I #14 May 28 2017 09:14:12

%S 0,1,1,4,1,2,1,27,4,2,1,5,1,2,2,256,1,5,1,5,2,2,1,28,4,2,27,5,1,3,1,

%T 3125,2,2,2,8,1,2,2,28,1,3,1,5,5,2,1,257,4,5,2,5,1,28,2,28,2,2,1,6,1,

%U 2,5,46656,2,3,1,5,2,3,1,31,1,2,5,5,2,3,1,257,256,2,1,6,2,2,2,28,1,6,2,5,2,2,2,3126,1,5,5,8,1,3,1,28,3

%N If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = sum_{p|n} b(p,n)^b(p,n).

%C a(1) = 0 (empty sum). - _Antti Karttunen_, May 28 2017

%H Antti Karttunen, <a href="/A112621/b112621.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 iff n is prime, a(n) = 2 iff n is a nonsquare semiprime (A006881). - _Robert G. Wilson v_, Dec 27 2005

%e 45 = 3^2 * 5^1. So a(45) = 2^2 + 1^1 = 5.

%t f[n_] := Block[{fi = Last@Transpose@FactorInteger@n}, Plus @@ (fi^fi)]; Rest@Array[f, 92] (* _Robert G. Wilson v_ *)

%o (PARI) A112621(n) = { my(f = factor(n), s = 0); for (k=1, #f~, s += (f[k, 2]^f[k, 2]); ); s; } \\ _Antti Karttunen_, May 28 2017

%Y Cf. A112622, A112623.

%K nonn

%O 1,4

%A _Leroy Quet_, Dec 25 2005

%E More terms from _Robert G. Wilson v_, Dec 27 2005

%E Term a(1) = 0 prepended, data section extended to 105 terms - _Antti Karttunen_, May 28 2017