OFFSET
1,2
COMMENTS
Inverse Moebius transform of A056671.
a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021
LINKS
FORMULA
a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
From Bernard Schott, May 03 2021: (Start)
a(p^k) = k+2 for p prime, or signature [k].
a(A006881(n)) = 9 for signature [1, 1].
a(A054753(n)) = 12 for signature [2, 1].
a(A065036(n)) = 15 for signature [3, 1].
a(A085986(n)) = 16 for signature [2, 2].
a(A178739(n)) = 18 for signature [4, 1].
a(A143610(n)) = 20 for signature [3, 2].
a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023
From Amiram Eldar, Sep 01 2023: (Start)
MATHEMATICA
a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]
PROG
(PARI) a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Ilya Gutkovskiy, Apr 15 2021
STATUS
approved