A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 3, 0, 0, 0, 7, 0, 4, 0, 5, 0, 0, 0, 9, 4, 0, 8, 7, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 11, 10, 0, 0, 21, 6, 8, 0, 13, 0, 16, 0, 21, 0, 0, 0, 15, 0, 0, 14, 31, 0, 0, 0, 17, 0, 0, 0, 37, 0, 0, 12, 19, 0, 0, 0, 35, 26, 0, 0, 21, 0, 0, 0, 33, 0, 20, 0, 23

Offset

1,8

Comments

Squarefree numbers are roots of a(n)=0 equation, while Min n for which a(n)=k is k^2. See also A000188, A008833.

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

Cf. A000188, A008833, A055653, A053570, A053571, A055631, A055632, A055653, A065465.

Sequence in context: A245203 A133949 A139808 * A170849 A292260 A322114

Adjacent sequences: A055651 A055652 A055653 * A055655 A055656 A055657

Keyword

nonn,easy

Author

Labos Elemer, Jun 07 2000

Status

approved