A167350 Totally multiplicative sequence with a(p) = (p+1)*(p-2) = p^2-p-2 for prime p.

1, 0, 4, 0, 18, 0, 40, 0, 16, 0, 108, 0, 154, 0, 72, 0, 270, 0, 340, 0, 160, 0, 504, 0, 324, 0, 64, 0, 810, 0, 928, 0, 432, 0, 720, 0, 1330, 0, 616, 0, 1638, 0, 1804, 0, 288, 0, 2160, 0, 1600, 0

Offset: 1

Author

Jaroslav Krizek, Nov 01 2009

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A003959, A166586.

Programs

Mathematica

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)

Formula

Multiplicative with a(p^e) = ((p+1)*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+1)*(p(k)-2))^e(k).

a(2k) = 0 for k >= 1.

a(n) = A003959(n) * A166586(n).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 3/p^3 + 2/p^4) = 0.1140434597... . - Amiram Eldar, Dec 15 2022