OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.
FORMULA
Multiplicative with a(p^e) = (2*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-1)^e(k).
Dirichlet g.f.: Product_{p prime} 1/(1 - 2*(p-1)*p^(-s)). - Robert Israel, May 19 2016
From Vaclav Kotesovec, Mar 08 2023: (Start)
Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - (2 - p^(2-s))/(p^s-2*p+2)).
Let f(s) = Product_{p prime} (1 - (2 - p^(2-s)) / (p^s - 2*p + 2)).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - 2/(p^2 - 2*p + 2)) = 0.353804459718477500968617797456682002952375753701841967763205003892191...,
f'(2) = f(2) * Sum_{p prime} 2*log(p) / ((p-1) * (p^2 - 2*p + 2)) = 0.350193097012820163529213089258238034020398107720137317340667886409682...
and gamma is the Euler-Mascheroni constant A001620. (End)
MAPLE
f:= proc(n) local f;
mul((2*(f[1]-1))^f[2], f = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, May 19 2016
MATHEMATICA
DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*2^(PrimeOmega[m]), {m, 1, 100}](* G. C. Greubel, May 19 2016, based on A003958 *)
f[p_, e_] := (2*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*(p-1)*X))[n], ", ")) \\ Vaclav Kotesovec, Mar 08 2023
CROSSREFS
KEYWORD
AUTHOR
Jaroslav Krizek, Oct 18 2009
STATUS
approved