A062790 Moebius transform of the cototient function A051953.

0, 1, 1, 1, 1, 2, 1, 2, 2, 4, 1, 3, 1, 6, 5, 4, 1, 6, 1, 5, 7, 10, 1, 6, 4, 12, 6, 7, 1, 8, 1, 8, 11, 16, 9, 8, 1, 18, 13, 10, 1, 12, 1, 11, 12, 22, 1, 12, 6, 20, 17, 13, 1, 18, 13, 14, 19, 28, 1, 13, 1, 30, 16, 16, 15, 20, 1, 17, 23, 24, 1, 16, 1, 36, 24, 19, 15, 24, 1, 20, 18, 40, 1, 19

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 1 .. 2000 from Harry J. Smith)

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Formula

a(n) = Sum f(n/d)*mu(d), where d divides n and f(x) = x-phi(x) = A051953(x).

a(n) = A056239(A318836(n)). - Antti Karttunen, Nov 24 2018

From Amiram Eldar, Dec 15 2023: (Start)

a(n) = A000010(n) - A007431(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 6/Pi^2 - 36/Pi^4. (End)

Example

n = 255, its divisors are {1,3,5,25,17,51,85,255}, A051953(255/d) = {127,21,19,1,7,1,1,0}, mu(d) = {1,-1,-1,1,-1,1,1,-1}, the sum is a(255) = 127-21-19+1-7+1+1+0 = 130-47 = 83.

Mathematica

Table[DirichletConvolve[MoebiusMu[n], n-EulerPhi[n], n, k], {k, 100}]  (* Amiram Eldar, Nov 24 2018 *)

Prog

(PARI) A062790(n)={
    local(a=0) ;
    fordiv(n, d,
        a += moebius(d)*(n/d-eulerphi(n/d)) ;
    ) ;
    return(a) ;
} \\ R. J. Mathar, Mar 24 2012
(PARI) A062790(n) = sumdiv(n, d, moebius(n/d)*(d-eulerphi(d))); \\ Antti Karttunen, Nov 24 2018

Crossrefs

Cf. A000010, A001065, A007431, A051953, A056239, A318836.

Sequence in context: A365461 A131097 A218666 * A046640 A347101 A240388

Adjacent sequences: A062787 A062788 A062789 * A062791 A062792 A062793

Keyword

nonn

Author

Labos Elemer, Jul 19 2001

Extensions

OFFSET changed from 0 to 1 by Harry J. Smith, Aug 11 2009

Status

approved