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
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