A167355 Totally multiplicative sequence with a(p) = (p-2)*(p+2) = p^2-4 for prime p.

1, 0, 5, 0, 21, 0, 45, 0, 25, 0, 117, 0, 165, 0, 105, 0, 285, 0, 357, 0, 225, 0, 525, 0, 441, 0, 125, 0, 837, 0, 957, 0, 585, 0, 945, 0, 1365, 0, 825, 0, 1677, 0, 1845, 0, 525, 0, 2205, 0, 2025, 0, 1425, 0, 2805, 0, 2457, 0, 1785, 0, 3477, 0, 3717, 0, 1125

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. A166586, A166590.

Programs

Mathematica

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^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 11 2016 *)

Formula

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

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

a(n) = A166586(n) * A166590(n).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 1/p^2 + 4/p^3 + 4/p^4) = 0.128353657048... . - Amiram Eldar, Dec 15 2022