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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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,changed

AUTHOR

Michael Somos, Aug 22 2008

STATUS

approved

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Last modified December 1 21:09 EST 2022. Contains 358484 sequences. (Running on oeis4.)