A167352 Totally multiplicative sequence with a(p) = (p+1)*(p-3) = p^2-2p-3 for prime p.

1, -3, 0, 9, 12, 0, 32, -27, 0, -36, 96, 0, 140, -96, 0, 81, 252, 0, 320, 108, 0, -288, 480, 0, 144, -420, 0, 288, 780, 0, 896, -243, 0, -756, 384, 0, 1292, -960, 0, -324, 1596, 0, 1760, 864, 0, -1440, 2112, 0, 1024, -432, 0, 1260, 2700, 0, 1152, -864, 0

Offset: 1

Author

Jaroslav Krizek, Nov 01 2009

Keywords

• sign
• mult

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003959, A166589.

Programs

Mathematica

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p+1)*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+1)*(p(k)-3))^e(k).

a(3k) = 0 for k >= 1.

a(n) = A003959(n) * A166589(n).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 1/p^2 + 5/p^3 + 3/p^4) = 0.0629795941629... . - Amiram Eldar, Dec 15 2022