OFFSET
1,8
COMMENTS
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n - Sum_{u|n, gcd(u,n/u) = 1} phi(u), i.e. when u is a unitary divisor of n.
a(n) = n - A055653(n). - Sean A. Irvine, Mar 30 2022
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065465 = 0.11848616... . - Amiram Eldar, Oct 04 2024
MATHEMATICA
Table[n - DivisorSum[n, EulerPhi[#] &, CoprimeQ[#, n/#] &], {n, 92}] (* Michael De Vlieger, Oct 26 2017 *)
f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 0; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 04 2024 *)
PROG
(Haskell)
a055654 n = a055654_list !! (n-1)
a055654_list = zipWith (-) [1..] a055653_list
-- Reinhard Zumkeller, Mar 11 2012
(PARI) a(n) = n - sumdiv(n, d, if (gcd(d, n/d)==1, eulerphi(d))); \\ Michel Marcus, Oct 27 2017
(PARI) a(n) = {my(f = factor(n)); n - prod(k = 1, #f~, f[k, 1]^f[k, 2] - f[k, 1]^(f[k, 2] - 1) + 1); } \\ Amiram Eldar, Oct 04 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Jun 07 2000
STATUS
approved