A360908 Multiplicative with a(p^e) = 2*e - 1.

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, 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, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 7, 7, 1, 1, 3, 1, 1

Offset: 1

Author

Vaclav Kotesovec, Feb 25 2023

Keywords

• nonn
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A048691, A322327, A361430, A367822.

Programs

Mathematica

a[n_] := Times @@ ((2*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, (1+(1+1/X)/(1-1/X)^2))[n], ", "))

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

Formula

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/(p^s*(p^s-1))).

Sum_{k=1..n} a(k) ~ c*n, where c = A367822 = Product_{p prime} (1 + 2/(p*(p-1))) = 3.279577150984783607372919498914633983999130708105267540952619534539808381...

a(n) = A361430(n^2). - Amiram Eldar, Feb 11 2024