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A345301 a(n) = Sum_{p|n, p prime} p^pi(n/p).

%I A345301 #15 Jan 22 2025 17:32:44
%S A345301 0,1,1,2,1,7,1,4,9,13,1,17,1,23,52,16,1,43,1,41,130,43,1,113,125,77,
%T A345301 81,113,1,270,1,64,364,145,968,371,1,275,898,881,1,1328,1,377,1354,
%U A345301 535,1,1241,2401,1137,2476,681,1,2699,4456,2913,6922,1053,1,10710,1,2079,8962
%N A345301 a(n) = Sum_{p|n, p prime} p^pi(n/p).
%H A345301 Antti Karttunen, <a href="/A345301/b345301.txt">Table of n, a(n) for n = 1..10000</a>
%F A345301 a(p^k) = p^pi(p^(k-1)), for p prime and k >= 1. - _Wesley Ivan Hurt_, Jun 26 2024
%e A345301 a(12) = Sum_{p|12} p^pi(12/p) = 2^pi(6) + 3^pi(4) = 2^3 + 3^2 = 17.
%t A345301 Table[Sum[k^PrimePi[n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
%o A345301 (Python)
%o A345301 from sympy import primefactors, primepi
%o A345301 def A345301(n): return sum(p**primepi(n//p) for p in primefactors(n)) # _Chai Wah Wu_, Jun 13 2021
%o A345301 (PARI) A345301(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 1]^primepi(n/f[i, 1]))); \\ _Antti Karttunen_, Jan 22 2025
%Y A345301 Cf. A000720.
%K A345301 nonn
%O A345301 1,4
%A A345301 _Wesley Ivan Hurt_, Jun 13 2021