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A343525 If n = Product (p_j^k_j) then a(n) = Product (2*p_j^k_j + 1), with a(1) = 1.

%I A343525 #16 Jul 24 2024 09:20:48
%S A343525 1,5,7,9,11,35,15,17,19,55,23,63,27,75,77,33,35,95,39,99,105,115,47,
%T A343525 119,51,135,55,135,59,385,63,65,161,175,165,171,75,195,189,187,83,525,
%U A343525 87,207,209,235,95,231,99,255,245,243,107,275,253,255,273,295,119,693,123,315,285,129,297,805
%N A343525 If n = Product (p_j^k_j) then a(n) = Product (2*p_j^k_j + 1), with a(1) = 1.
%C A343525 The unitary analog of A060640.
%H A343525 Alois P. Heinz, <a href="/A343525/b343525.txt">Table of n, a(n) for n = 1..20000</a>
%F A343525 a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * usigma(n/d).
%F A343525 a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d).
%F A343525 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-1) - 2/p^(2*s-1)). - _Amiram Eldar_, Jul 24 2024
%p A343525 a:= n-> mul(2*i[1]^i[2]+1, i=ifactors(n)[2]):
%p A343525 seq(a(n), n=1..80);  # _Alois P. Heinz_, Apr 18 2021
%t A343525 a[1] = 1; a[n_] := Times @@ ((2 #[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 66}]
%o A343525 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2*f[k,1]^f[k,2]+1; f[k,2]=1); factorback(f); \\ _Michel Marcus_, Apr 18 2021
%Y A343525 Cf. A001221, A034444, A034448, A060640, A107759, A145388, A298473.
%K A343525 nonn,look,mult
%O A343525 1,2
%A A343525 _Ilya Gutkovskiy_, Apr 18 2021