A275257 -id:A275257 - cp's OEIS frontend

A077614 Let F(m,k) be the number of positive integers j <= m such that gcd(j,k) = 1; then a(n) is the number of positive integers k such that F(n,k) = k.

1, 0, 1, 2, 0, 1, 1, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 0, 1, 3, 3, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 0, 2, 3, 2, 1, 1, 0, 0, 1, 0, 2, 2, 1, 1, 1, 0, 1, 0, 0, 2, 3, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 2, 2, 1, 2, 2, 1, 0, 1, 0, 1, 0

Offset: 1

Christian G. Bower and Leroy Quet, Nov 11 2002

nonn

Cf. A077615 (least inverse), A275257 (the F array).

Name edited by Jon E. Schoenfield, Oct 27 2019

A275258 Toth's partial sum over the number of divisors of the greatest unitary divisor.

Original entry on oeis.org

1, 3, 4, 6, 6, 11, 8, 11, 11, 16, 12, 21, 14, 21, 23, 20, 18, 29, 20, 32, 30, 31, 24, 39, 27, 36, 30, 42, 30, 57, 32, 37, 45, 46, 47, 56, 38, 51, 52, 59, 42, 77, 44, 62, 63, 61, 48, 71, 51, 69, 67, 72, 54, 77, 70, 78, 74, 76, 60, 113, 62, 81, 83

Offset: 1

R. J. Mathar, Jul 21 2016

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009), Article 09.5.2, function S**(n).

Cf. A000005, A065486, A165430.

A275258 := proc(n)
    local a,d ;
    a := 0 ;
    for  d in A077610(n) do
        a := a+A005361(d)*A275257(n/d,d) ;
    end do:
    a ;
end proc:
seq(A275258(n),n=1..80) ;

beta[n_] := Times @@ Transpose[FactorInteger[n]][[2]]; phi[x_, n_] := Sum[Boole[ GCD[k, n] == 1 ], {k, 1, x}]; a[n_] := DivisorSum[n, beta[#] * phi[n/#, #] &, GCD[#, n/#] == 1 &]; Array[a, 100] (* Amiram Eldar, Sep 22 2019 *)

a(n) = Sum_{k=1..n} A000005( A165430(n,k) ).

Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n * log(n)^2), where c = A065486. - Amiram Eldar, Dec 22 2023

cp's OEIS Frontend

A077614 Let F(m,k) be the number of positive integers j <= m such that gcd(j,k) = 1; then a(n) is the number of positive integers k such that F(n,k) = k.

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A275258 Toth's partial sum over the number of divisors of the greatest unitary divisor.

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