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A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

%I A304037 #10 May 09 2018 23:03:04
%S A304037 0,1,2,1,3,3,4,1,4,4,5,3,6,5,5,1,7,5,8,4,6,6,9,3,9,7,8,5,10,6,11,1,7,
%T A304037 8,7,5,12,9,8,4,13,7,14,6,7,10,15,3,16,10,9,7,16,9,8,5,10,11,17,6,18,
%U A304037 12,8,1,9,8,19,8,11,8,20,5,21,13,11,9,9,9,22,4,16,14,23,7,10,15,12,6
%N A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.
%H A304037 Ilya Gutkovskiy, <a href="/A304037/a304037.pdf">Extended graphical example</a>
%H A304037 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A304037 If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
%F A304037 a(p^k) = A000720(p)^k where p is a prime.
%F A304037 a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
%F A304037 As an example:
%F A304037 a(A000040(k)) = k.
%F A304037 a(A006450(k)) = A000040(k).
%F A304037 a(A038580(k)) = A006450(k).
%F A304037 a(A001248(k)) = a(A011757(k)) = A000290(k).
%F A304037 a(A030078(k)) = a(A055875(k)) = A000578(k).
%F A304037 a(A002110(k)) = a(A011756(k)) = A000217(k).
%F A304037 a(A061742(k)) = A000330(k).
%F A304037 a(A115964(k)) = A000537(k).
%F A304037 a(A080696(k)) = A007504(k).
%F A304037 a(A076954(k)) = A001923(k).
%e A304037 a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.
%t A304037 a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]
%Y A304037 Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.
%K A304037 nonn
%O A304037 1,3
%A A304037 _Ilya Gutkovskiy_, May 05 2018