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%I #20 Mar 03 2023 17:26:57
%S 0,2,3,6,5,17,7,14,12,27,11,45,13,37,38,30,17,62,19,71,52,57,23,101,
%T 30,67,39,97,29,162,31,62,80,87,82,162,37,97,94,159,41,220,43,149,137,
%U 117,47,213,56,152,122,175,53,197,126,217,136,147,59,410,61,157,187,126,148,336,67,227,164,342,71,362,73,187,213,253,172,394,79
%N a(n) = Sum_{d|n} d*omega(d).
%H Antti Karttunen, <a href="/A329354/b329354.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A329354/a329354.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F a(n) = Sum_{d|n} d*A001221(d).
%F a(n) = A180253(n) - A323599(n).
%F a(n) = A328260(n) + A329375(n).
%F a(n) = Sum_{d|n} (n/d) * sopf(d). - _Wesley Ivan Hurt_, May 24 2021
%F Dirichlet g.f.: primezeta(s-1) * zeta(s-1) * zeta(s). - _Ilya Gutkovskiy_, Aug 18 2021
%F Conjecture: Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (log(log(n)) + A077761) / 12. - _Vaclav Kotesovec_, Mar 03 2023
%t Table[Sum[d*PrimeNu[d], {d, Divisors[n]}], {n, 1, 100}] (* _Vaclav Kotesovec_, Aug 18 2021 *)
%o (PARI) A329354(n) = sumdiv(n,d,omega(d)*d);
%Y Cf. A001221, A013939, A062799, A180253, A323599, A328260, A329375.
%K nonn
%O 1,2
%A _Antti Karttunen_, Nov 15 2019
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