A182139 Inverse Moebius transform of A061142.

1, 3, 3, 7, 3, 9, 3, 15, 7, 9, 3, 21, 3, 9, 9, 31, 3, 21, 3, 21, 9, 9, 3, 45, 7, 9, 15, 21, 3, 27, 3, 63, 9, 9, 9, 49, 3, 9, 9, 45, 3, 27, 3, 21, 21, 9, 3, 93, 7, 21, 9, 21, 3, 45, 9, 45, 9, 9, 3, 63, 3, 9, 21, 127, 9, 27, 3, 21, 9, 27, 3, 105, 3, 9, 21, 21, 9

Offset

1,2

Comments

a(n) is multiplicative with a(p^e) = -1 + 2^(e+1).

If s is squarefree then a(s) = A048691(s).

More generally: Let a_q(n) be multiplicative with a_q(p^e) = (q^(e+1)-1)/ (q-1) for prime p, e >= 0 and some fixed integer q. Then a_q(n) is the inverse Moebius transform of the completely multiplicative sequence b_q(n) = q^bigomega(n) with b_q(p) = q and b_q(1) = 1. For q = 1 see a_q(n) = A000005(n) and b_q(n) = A000012(n), for q = 0 see a_q(n) = A000012(n) and b_q(n) = A000007(n) with offset 1, and for q = -1 see a_q(n) = A010052(n) with offset 1 and b_q(n) = A008836(n). - Werner Schulte, Feb 20 2019

Links

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Formula

a(n) = Sum_{d|n} 2^Omega(d) = Sum_{d|n} 2^A001222(d) = Sum_{d|n} A061142(d).

Dirichlet g.f.: zeta(s)^3 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2).

Example

a(12) = a(2^2 * 3^1) = (-1 + 2^(2+1)) * (-1 + 2^(1+1)) = 7 * 3 = 21; or, using the divisors set {1,2,3,4,6,12}: 2^0 + 2^1 + 2^1 + 2^2 + 2^2 + 2^3 = 21.

Mathematica

t[n_] := DivisorSum[n, 2^PrimeOmega[#]&]; Table[t[n], {n, 100}]

Prog

(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)/(1 - 2*X))[n], ", ")) \\ Vaclav Kotesovec, Mar 14 2023

Crossrefs

Cf. A001222, A061142, A048691.

Sequence in context: A034257 A145501 A370296 * A092693 A134661 A135434

Adjacent sequences: A182136 A182137 A182138 * A182140 A182141 A182142

Keyword

nonn,mult

Author

Enrique Pérez Herrero, Apr 14 2012

Status

approved