OFFSET
1,2
COMMENTS
Dirichlet convolution of A001157 and A000012. Dirichlet convolution of A000005 and A000290 (Jovovic formula). - R. J. Mathar, Feb 03 2011
Sum of the squares of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms
FORMULA
a(n) = Sum_{d|n} d^2*tau(n/d). - Vladeta Jovovic, Jul 31 2002
Equals A134577 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 02 2007
G.f.: Sum_{k>=1} sigma_2(k)*x^k/(1 - x^k), where sigma_2(k) is the sum of squares of divisors of k (A001157). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s-2)*zeta(s)^2. - Benedict W. J. Irwin, Jul 14 2018
a(n) is multiplicative with a(p^e) = (p^(2*e + 4) - (e+2) * p^2 + e+1)) / (p^2 - 1)^2. - Michael Somos, Jul 15 2018
Sum_{k=1..n} a(k) ~ Zeta(3)^2 * n^3 / 3. - Vaclav Kotesovec, Nov 04 2018
EXAMPLE
G.f. = x + 6*x^2 + 11*x^3 + 27*x^4 + 27*x^5 + 66*x^6 + 51*x^7 + 112*x^8 + 102*x^9 + ... - Michael Somos, Jul 15 2018
MATHEMATICA
a[n_] := Plus @@ DivisorSigma[2, Divisors[n]]; Array[a, 41] (* Robert G. Wilson v, May 05 2010 *)
a[ n_] := If[ n < 1, 0, Times @@ (If[ # == 1, 1, (#^(2 #2 + 4) - (#2 + 2) #^2 + #2 + 1) / (#^2 - 1)^2] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 15 2018 *)
PROG
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
EXTENSIONS
a(38) corrected by Ilya Gutkovskiy, Jan 16 2016
STATUS
approved