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

1,5

Links

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

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

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 *)

Prog

(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

Crossrefs

Cf. A000244 (powers of 3).

Sequence in context: A234020 A348259 A276833 * A122274 A373023 A340525

Adjacent sequences: A166583 A166584 A166585 * A166587 A166588 A166589

Keyword

nonn,mult

Author

Jaroslav Krizek, Oct 17 2009

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

Status

approved