A095001 - OEIS

%I #12 Sep 07 2020 06:30:33

%S 1,3,4,7,6,24,8,23,31,68,12,196,14,192,384,279,18,1473,20,1792,2552,

%T 2192,24,12068,3131,8388,19714,19124,30,116474,32,65815,178512,131396,

%U 94968,841093,38,524688,1596560,1450368,42,7280934,44,4211500,16305666

%N a(n) = Sum_{d|n} rad(d)^(n/d), where rad(d) = A007947(d) is the squarefree kernel of d.

%H Amiram Eldar, <a href="/A095001/b095001.txt">Table of n, a(n) for n = 1..6287</a>

%F G.f. satisfies: A(x) = Sum_{n>=1} [1/(1-A007947(n)*x^n) - 1].

%F a(n) = Sum_{d|n} A007947(d)^(n/d); a(p) = p+1, for prime p.

%t rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; a[n_] := DivisorSum[n, rad[#]^(n / #) &]; Array[a, 50] (* _Amiram Eldar_, Sep 07 2020 *)

%o (PARI) {a(n)=sumdiv(n,d,prod(i=1,omega(d),factor(d)[i,1])^(n/d))} /* also formed by the log of G094947(x): */ {a(n)=n*polcoeff(sum(k=1,n,-log(1-prod(i=1,omega(k),factor(k)[i,1])*x^k+x*O(x^n))/k),n)}

%Y Cf. A007947, A094947.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 26 2004