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

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

Author

R. J. Mathar, Jul 21 2016

Keywords

• nonn
• look
• easy

Links

• 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).

Crossrefs

Cf. A000005, A065486, A165430.

Programs

Maple

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) ;

Mathematica

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 *)

Formula

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