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A309153
a(n) = A000203(n)*A001227(n).
1
1, 3, 8, 7, 12, 24, 16, 15, 39, 36, 24, 56, 28, 48, 96, 31, 36, 117, 40, 84, 128, 72, 48, 120, 93, 84, 160, 112, 60, 288, 64, 63, 192, 108, 192, 273, 76, 120, 224, 180, 84, 384, 88, 168, 468, 144, 96, 248, 171, 279, 288, 196, 108, 480, 288, 240, 320, 180, 120, 672, 124, 192, 624, 127, 336, 576, 136, 252, 384
OFFSET
1,2
COMMENTS
A001227(n) is denoted by Delta_0(n) in Glaisher 1907.
a(n) = A000203(n) iff n is a power of 2.
FORMULA
a(n) = sigma(n)*delta(n).
Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - Amiram Eldar, Nov 01 2022
From Amiram Eldar, Dec 04 2023: (Start)
Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)
MATHEMATICA
Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* Michael De Vlieger, Nov 22 2019 *)
f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)
KEYWORD
nonn,mult,easy
AUTHOR
Omar E. Pol, Jul 14 2019
STATUS
approved