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

Author

Enrique Pérez Herrero, Apr 14 2012

Keywords

• nonn
• mult

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A001222, A061142, A048691.

Programs

Mathematica

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

PARI

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

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).