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%I #30 Mar 24 2023 08:22:16
%S 1,2,3,6,5,6,7,12,12,10,11,18,13,14,15,28,17,24,19,30,21,22,23,36,30,
%T 26,36,42,29,30,31,56,33,34,35,72,37,38,39,60,41,42,43,66,60,46,47,84,
%U 56,60,51,78,53,72,55,84,57,58,59,90,61,62,84,120,65,66,67
%N Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).
%H Antti Karttunen, <a href="/A340774/b340774.txt">Table of n, a(n) for n = 1..16384</a>
%F Multiplicative with a(p^e) = (p^(e+1)-p^floor((e+1)/2))/(p-1).
%F Dirichlet convolution of A000010 and A069290.
%F Dirichlet convolution with A055615 equals A037213.
%F G.f.: Sum_{k>=1} k * x^(k^2) / (1 - x^(k^2))^2. - _Ilya Gutkovskiy_, Aug 19 2021
%F Sum_{k=1..n} a(k) ~ zeta(3)*n^2/2. - _Vaclav Kotesovec_, Aug 19 2021
%F a(n) = n * Sum_{d^2|n} 1/d. - _Wesley Ivan Hurt_, Feb 14 2022
%p a:= n-> mul((i[1]^(i[2]+1)-i[1]^iquo(i[2]+1, 2))/(i[1]-1), i=ifactors(n)[2]):
%p seq(a(n), n=1..77); # _Alois P. Heinz_, Jan 20 2021
%t f[p_, e_] := (p^(e + 1) - p^Floor[(e + 1)/2])/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 20 2021 *)
%o (PARI) A340774(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((p^(e+1)-(p^((e+1)\2))) / (p-1))); }; \\ _Antti Karttunen_, Aug 19 2021
%Y Cf. A000010, A069290, A055615, A037213.
%Y Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k =
%Y 0..10: A046951 (k=0), this sequence (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
%K nonn,easy,mult
%O 1,2
%A _Werner Schulte_, Jan 20 2021
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