A166589 Totally multiplicative sequence with a(p) = p-3 for prime p.

1, -1, 0, 1, 2, 0, 4, -1, 0, -2, 8, 0, 10, -4, 0, 1, 14, 0, 16, 2, 0, -8, 20, 0, 4, -10, 0, 4, 26, 0, 28, -1, 0, -14, 8, 0, 34, -16, 0, -2, 38, 0, 40, 8, 0, -20, 44, 0, 16, -4, 0, 10, 50, 0, 16, -4, 0, -26, 56, 0, 58, -28, 0, 1, 20, 0, 64, 14, 0, -8, 68, 0

Offset: 1

Author

Jaroslav Krizek, Oct 17 2009

Keywords

• sign
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A166586.

Programs

Mathematica

a[1] = 1; a[p_?PrimeQ] := p-3; a[n_] := Times @@ Power @@@ ({#[[1]]-3, #[[2]]}& /@ FactorInteger[n]); Array[a, 72] (* Jean-François Alcover, Jul 19 2017 *)

PARI

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] -=3); factorback(f); \\ Michel Marcus, Jun 09 2014

Formula

Multiplicative with a(p^e) = (p-3)^e. If n = Product p(k)^e(k) then a(n) = Product (p(k)-3)^e(k). a(3k) = 0 for k >= 1. Abs (a(2^k)) = 1 for k >= 1.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p*(p-1)/(p^2-p+3)) = 0.196347937547... . - Amiram Eldar, Jan 20 2024

Extensions

More terms from Michel Marcus, Jun 09 2014