A275254 The bi-unitary gcd-sum function.

1, 3, 5, 7, 9, 14, 13, 15, 17, 25, 21, 30, 25, 36, 43, 31, 33, 47, 37, 57, 61, 58, 45, 64, 49, 69, 53, 82, 57, 108, 61, 63, 99, 91, 113, 99, 73, 102, 117, 117, 81, 163, 85, 132, 141, 124, 93, 130, 97, 135, 155, 157, 105, 146, 181

Offset

1,2

Comments

Row sums of A165430.

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 P**(n).

Formula

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

Sum_{k=1..n} a(k) = c * n^2 * log(n) / 2 + O(n^2), where c = Product_{p prime} (1 - (3*p-1)/(p^2*(p+1))) = zeta(2) * Product_{p prime} (1 - (2*p-1)^2/p^4) = A013661 * A256392 = 0.35823163000196141456... . - Amiram Eldar, Dec 22 2023

Maple

Pstarstar := proc(n)
    add(A165430(k, n), k=1..n) ;
end proc:

Mathematica

phi[x_, n_] := Sum[Boole[GCD[k, n] == 1], {k, 1, x}]; uphi[1]=1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); a[n_] := DivisorSum[n, uphi[#] * phi[n/#, #] &, GCD[#, n/#] == 1 &]; Array[a, 100] (* Amiram Eldar, Sep 09 2019 *)

Crossrefs

Cf. A013661, A165430, A256392.

Sequence in context: A107220 A249412 A098758 * A029608 A211135 A145388

Adjacent sequences: A275251 A275252 A275253 * A275255 A275256 A275257

Keyword

nonn

Author

R. J. Mathar, Jul 21 2016

Status

approved