OFFSET
1,3
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
a(n) is multiplicative with a(2^e) = (e-1) * 2^e if e>0, a(p^e) = (e+1) * p^e if p>2.
a(4*n + 2) = 0.
G.f.: Sum_{k>0} k * x^k / (1 - (-x)^k)^2.
From Amiram Eldar, Nov 29 2022: (Start)
a(n) = n * A112329(n).
Dirichlet g.f.: zeta(s-1)^2*(1+2^(3-2*s)-2^(2-s)).
Sum_{k=1..n} a(k) ~ n^2*log(n)/4 + (4*gamma-1)*n^2/8, where gamma is Euler's constant (A001620). (End)
EXAMPLE
q + 6*q^3 + 4*q^4 + 10*q^5 + 14*q^7 + 16*q^8 + 27*q^9 + 22*q^11 + 24*q^12 + ...
MATHEMATICA
Abs@Total[# (-1)^Divisors[#]] & /@ Range[68] (* George Beck, Oct 25 2014 *)
f[p_, e_] := (e + 1)*p^e; f[2, e_] := (e - 1)*2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)
PROG
(PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; (e - (-1)^p) * p^e)))}
(PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k * x^k / (1 - (-x)^k)^2, x*O(x^n)), n))}
(Haskell)
a143520 n = product $ zipWith (\p e -> (e + 2 * mod p 2 - 1) * p ^ e)
(a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jan 21 2014
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Aug 22 2008
STATUS
approved