A143520 a(n) is n times number of divisors of n if n is odd, zero if n is twice odd, n times number of divisors of n/4 if n is divisible by 4.

1, 0, 6, 4, 10, 0, 14, 16, 27, 0, 22, 24, 26, 0, 60, 48, 34, 0, 38, 40, 84, 0, 46, 96, 75, 0, 108, 56, 58, 0, 62, 128, 132, 0, 140, 108, 74, 0, 156, 160, 82, 0, 86, 88, 270, 0, 94, 288, 147, 0, 204, 104, 106, 0, 220, 224, 228, 0, 118, 240, 122, 0, 378, 320, 260, 0, 134, 136

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.

A038040(2*n + 1) = a(2*n + 1); 4 * A038040(n) = a(4*n).

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

Cf. A001620, A027748, A038040, A112329, A124010.

Sequence in context: A177898 A082209 A264770 * A075450 A145979 A182164

Adjacent sequences: A143517 A143518 A143519 * A143521 A143522 A143523

Keyword

nonn,mult

Author

Michael Somos, Aug 22 2008

Status

approved