A166586 Totally multiplicative sequence with a(p) = p - 2 for prime p.

1, 0, 1, 0, 3, 0, 5, 0, 1, 0, 9, 0, 11, 0, 3, 0, 15, 0, 17, 0, 5, 0, 21, 0, 9, 0, 1, 0, 27, 0, 29, 0, 9, 0, 15, 0, 35, 0, 11, 0, 39, 0, 41, 0, 3, 0, 45, 0, 25, 0, 15, 0, 51, 0, 27, 0, 17, 0, 57, 0, 59, 0, 5, 0, 33, 0, 65, 0, 21, 0, 69, 0, 71, 0, 9, 0

Offset: 1

Author

Jaroslav Krizek, Oct 17 2009

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000244 (powers of 3).

Programs

Maple

f:= proc(n) local t;
    mul((t[1]-2)^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jun 07 2016

Mathematica

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 06 2016 *)

PARI

a(n) = my(f = factor(n)); for (i=1, #f~, f[i,1] -= 2); factorback(f); \\ Michel Marcus, Dec 13 2014

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X+2*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2023

Formula

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

a(A000244(n)) = 1. - Michel Marcus, Dec 13 2014

Dirichlet g.f.: 1 / Product_{p prime} (1 - p^(1 - s) + 2*p^(-s)). The Dirichlet inverse is multiplicative with b(p) = 2 - p, b(p^e) = 0, for e > 1. - Álvar Ibeas, Nov 24 2017 [corrected by Vaclav Kotesovec, Feb 10 2023]

Sum_{k=1..n} a(k) ~ c * n^2/2, where c = Product_{primes} (1 - 1/(1 + p*(p-1)/2)) = 0.3049173579282080265466051390930446635010608835584906520231313997... - Vaclav Kotesovec, Feb 10 2023

Extensions

More terms from Alonso del Arte, Dec 10 2014

a(69) and a(75) corrected by G. C. Greubel, Jun 06 2016

Erroneous formula and program removed by G. C. Greubel, Jun 06 2016