OFFSET
1,2
COMMENTS
n divides a(n) for n: 1, 3, 4, 8, 12, 24, 28, 84, 88, 144, 264, 432, 440, 476, 1320, ...
Inverse Möbius transform of A064840. - Antti Karttunen, Mar 28 2019
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(p) = 2p + 3 for p = primes (A000040).
a(n) = Sum_{d|n} A064840(d).
Multiplicative with a(p^e) = Sum_{k=0..e} (p^(k+1)-1)*(k+1)/(p-1). - Aloe Poliszuk, Oct 31 2025
From Amiram Eldar, Oct 31 2025: (Start)
Dirichlet g.f.: zeta(s)^3 * zeta(s-1)^2 / zeta(2*s-1).
Sum_{k=1..n} a(k) ~ (Pi^6/(432*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 3*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)
EXAMPLE
a(6) = tau(1)*sigma(1) + tau(2)*sigma(2) + tau(3)*sigma(3) + tau(6)*sigma(6) = (1*1) + (2*3) + (2*4) + (4*12) = 63.
MAPLE
f:= proc(n) local d; add(numtheory:-tau(d)*numtheory:-sigma(d), d = numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Oct 30 2025
MATHEMATICA
Table[Sum[DivisorSigma[0, k]*DivisorSigma[1, k], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Mar 23 2019 *)
PROG
(Magma) [&+[NumberOfDivisors(d) * SumOfDivisors(d): d in Divisors(n)]: n in [1..100]];
(PARI) a(n) = my(d=divisors(n)); sum(i=1, #d, numdiv(d[i])*sigma(d[i])) \\ Felix Fröhlich, Mar 23 2019
(PARI) a(n) = sumdiv(n, d, numdiv(d)*sigma(d)); \\ Michel Marcus, Mar 24 2019
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Jaroslav Krizek, Mar 23 2019
STATUS
approved
