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A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

%I A304117 #12 Jul 24 2018 19:33:40
%S A304117 1,1,2,2,3,2,4,3,4,3,5,4,6,4,6,4,7,4,8,6,8,5,9,6,6,6,6,8,10,6,11,5,10,
%T A304117 7,12,8,12,8,12,9,13,8,14,10,12,9,15,8,8,6,14,12,16,6,15,12,16,10,17,
%U A304117 12,18,11,16,6,18,10,19,14,18,12,20,12,21,12,12,16,20,12,22,12
%N A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.
%H A304117 Andrew Howroyd, <a href="/A304117/b304117.txt">Table of n, a(n) for n = 1..1000</a>
%H A304117 Ilya Gutkovskiy, <a href="/A304117/a304117.pdf">Extended graphical example</a>
%H A304117 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A304117 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A304117 a(n) = A005361(n)*A156061(n).
%F A304117 a(p^k) = A000720(p)*k where p is a prime.
%F A304117 a(A002110(m)^k) = k^m*m!.
%F A304117 As an example:
%F A304117 a(A000040(k)) = k.
%F A304117 a(A006450(k)) = A000040(k).
%F A304117 a(A001248(k)) = a(A031215(k)) = A005843(k).
%F A304117 a(A030078(k)) = a(A031336(k)) = A008585(k)
%F A304117 a(A061742(k)) = A000165(k).
%F A304117 a(A115964(k)) = A032031(k).
%F A304117 a(A002110(k)) = A000142(k).
%F A304117 a(A080696(k)) = A002110(k).
%e A304117 a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.
%t A304117 a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]
%o A304117 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1])*f[k,2]; f[k, 2] = 1); factorback(f); \\ _Michel Marcus_, May 06 2018
%Y A304117 Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.
%K A304117 nonn,mult
%O A304117 1,3
%A A304117 _Ilya Gutkovskiy_, May 06 2018