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A345355 a(n) = Sum_{p|n, p prime} p^omega(n/p).

%I A345355 #21 Apr 13 2025 18:34:57
%S A345355 0,1,1,2,1,5,1,2,3,7,1,7,1,9,8,2,1,11,1,9,10,13,1,7,5,15,3,11,1,38,1,
%T A345355 2,14,19,12,13,1,21,16,9,1,62,1,15,14,25,1,7,7,27,20,17,1,11,16,11,22,
%U A345355 31,1,42,1,33,16,2,18,134,1,21,26,78,1,13,1,39,28,23,18,182
%N A345355 a(n) = Sum_{p|n, p prime} p^omega(n/p).
%H A345355 Antti Karttunen, <a href="/A345355/b345355.txt">Table of n, a(n) for n = 1..20000</a>
%F A345355 a(p) = 1 for p prime.
%F A345355 a(n) = Sum_{d|n} d^omega(n/d) * c(d), where c = A010051. - _Wesley Ivan Hurt_, Apr 13 2025
%e A345355 a(30) = Sum_{p|30} p^omega(30/p) = 2^omega(15) + 3^omega(10) + 5^omega(6) = 2^2 + 3^2 + 5^2 = 38.
%t A345355 Table[Sum[k^PrimeNu[n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
%o A345355 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^omega(n/f[k,1])); \\ _Michel Marcus_, Jun 16 2021
%Y A345355 Cf. A001221 (omega), A010051, A369744.
%Y A345355 Cf. also A369741, A369905, A369907.
%K A345355 nonn
%O A345355 1,4
%A A345355 _Wesley Ivan Hurt_, Jun 15 2021