A198302 a(n) = Sum_{d|n} d * sigma(n/d, d).

1, 5, 7, 21, 11, 65, 15, 133, 106, 245, 23, 1077, 27, 1041, 1637, 3365, 35, 9992, 39, 18361, 16401, 22841, 47, 134461, 15686, 106917, 179494, 355173, 59, 1220075, 63, 1593189, 1952705, 2228909, 631005, 13778268, 75, 9962313, 20732901, 34805473, 83, 113693883

Offset

1,2

Comments

Here sigma(n,k) is the sum of the k-th powers of the divisors of n.

Logarithmic derivative of A198301.

Links

Robert Israel, Table of n, a(n) for n = 1..6236

Formula

L.g.f.: Sum_{n>=1} Sum_{k>=1} sigma(n,k) * x^(n*k)/n.

Example

L.g.f.: L(x) = x + 5*x^2/2 + 7*x^3/3 + 21*x^4/4 + 11*x^5/5 + 65*x^6/6 + ...
Exponentiation yields the g.f. of A198301: exp(L(x)) = 1 + x + 3*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 42*x^6 + 62*x^7 + 131*x^8 + 206*x^9 + 398*x^10 + ... + A198301(n)*x^n + ...

Maple

f:= proc(n) local d; add(d*numtheory:-sigma[d](n/d), d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Nov 05 2024

Mathematica

a[n_] := DivisorSum[n, # * DivisorSigma[#, n/#] &]; Array[a, 40] (* Amiram Eldar, Aug 18 2023 *)

Prog

(PARI) {a(n)=sumdiv(n, d, d*sigma(n/d, d))}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=1, n\m, sigma(m, k)*x^(m*k)/m)+x*O(x^n)), n)}

Crossrefs

Cf. A000203, A198301 (exp), A185301, A198299.

Sequence in context: A258282 A192422 A120035 * A091154 A057424 A027152

Adjacent sequences: A198299 A198300 A198301 * A198303 A198304 A198305

Keyword

nonn,look

Author

Paul D. Hanna, Jan 27 2012

Status

approved