A167362 Totally multiplicative sequence with a(p) = (p-3)*(p+3) = p^2-9 for prime p.

1, -5, 0, 25, 16, 0, 40, -125, 0, -80, 112, 0, 160, -200, 0, 625, 280, 0, 352, 400, 0, -560, 520, 0, 256, -800, 0, 1000, 832, 0, 952, -3125, 0, -1400, 640, 0, 1360, -1760, 0, -2000, 1672, 0, 1840, 2800, 0, -2600, 2200, 0, 1600, -1280, 0, 4000, 2800, 0, 1792

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. A166589, A166591.

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

Formula

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

a(n) = A166589(n) * A166591(n).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 1/p^2 + 9/p^3 + 9/p^4) = 0.05980933853... . - Amiram Eldar, Dec 15 2022