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%I #16 Feb 11 2024 02:38:54
%S 1,1,1,3,1,1,1,5,3,1,1,3,1,1,1,7,1,3,1,3,1,1,1,5,3,1,5,3,1,1,1,9,1,1,
%T 1,9,1,1,1,5,1,1,1,3,3,1,1,7,3,3,1,3,1,5,1,5,1,1,1,3,1,1,3,11,1,1,1,3,
%U 1,1,1,15,1,1,3,3,1,1,1,7,7,1,1,3,1,1
%N Multiplicative with a(p^e) = 2*e - 1.
%H Vaclav Kotesovec, <a href="/A360908/b360908.txt">Table of n, a(n) for n = 1..10000</a>
%F Dirichlet g.f.: zeta(s) * Product_{p primes} (1 + 2/(p^s*(p^s-1)).
%F Sum_{k=1..n} a(k) ~ c*n, where c = Product_{p primes} (1 + 2/(p*(p-1)) = 3.279577150984783607372919498914633983999130708105267540952619534539808381...
%F a(n) = A361430(n^2). - _Amiram Eldar_, Feb 11 2024
%t a[n_] := Times @@ ((2*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Feb 25 2023 *)
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1+(1+1/X)/(1-1/X)^2))[n], ", "))
%o (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]=2*f[k,2]-1; f[k,2]=1); factorback(f); \\ _Michel Marcus_, Feb 25 2023
%Y Cf. A048691, A322327, A361430.
%K nonn,mult
%O 1,4
%A _Vaclav Kotesovec_, Feb 25 2023