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

Author

Labos Elemer, Jul 19 2001

Keywords

• nonn

Examples

			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.
		

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)

Crossrefs

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

Programs

Mathematica

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

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

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)

Extensions

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