%I #17 Nov 21 2022 10:52:35

%S 1,0,2,3,3,11,10,26,32,51,90,117,198,283,417,610,890,1284,1848,2615,

%T 3716,5217,7289,10222,14158,19514,26882,36805,50131,68428,92466,

%U 125128,168093,225775,302171,402876,536730,711601,942009,1243513,1638395,2152828,2823004

%N G.f.: exp( Sum_{n>=1} A005063(n)*x^n/n ), where A005063(n) = sum of squares of primes dividing n.

%C Euler transform of A061397. - _Peter Luschny_, Nov 21 2022

%H Paul D. Hanna, <a href="/A219224/b219224.txt">Table of n, a(n) for n = 0..1000</a>

%e G.f.: A(x) = 1 + 2*x^2 + 3*x^3 + 3*x^4 + 11*x^5 + 10*x^6 + 26*x^7 + 32*x^8 +...

%e where

%e log(A(x)) = 4*x^2/2 + 9*x^3/3 + 4*x^4/4 + 25*x^5/5 + 13*x^6/6 + 49*x^7/7 + 4*x^8/8 + 9*x^9/9 + 29*x^10/10 + 121*x^11/11 + 13*x^12/12 + 169*x^13/13 + 53*x^14/14 + 34*x^15/15 +...+ A005063(n)*x^n/n +...

%p # The function EulerTransform is defined in A358369.

%p a := EulerTransform(n -> ifelse(isprime(n), n, 0)):

%p seq(a(n), n = 0..42); # _Peter Luschny_, Nov 21 2022

%t a[n_] := SeriesCoefficient[ Exp[ Sum[ DivisorSum[k, Boole[PrimeQ[#]] * #^2&] * x^k/k, {k, 1, n+1}]], {x, 0, n}]; Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Jul 11 2017, from PARI *)

%o (PARI) {a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,isprime(d)*d^2)*x^k/k)+x*O(x^n)),n)}

%o for(n=0,50,print1(a(n),", "))

%Y Cf. A000607, A005063, A220427, A061397.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 15 2012