A275254 - OEIS

%I #15 Dec 22 2023 08:39:01

%S 1,3,5,7,9,14,13,15,17,25,21,30,25,36,43,31,33,47,37,57,61,58,45,64,

%T 49,69,53,82,57,108,61,63,99,91,113,99,73,102,117,117,81,163,85,132,

%U 141,124,93,130,97,135,155,157,105,146,181

%N The bi-unitary gcd-sum function.

%C Row sums of A165430.

%H Amiram Eldar, <a href="/A275254/b275254.txt">Table of n, a(n) for n = 1..10000</a>

%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Toth2/toth5.html">On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function</a>, JIS 12 (2009), Article 09.5.2, function P**(n).

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

%F 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

%p Pstarstar := proc(n)

%p     add(A165430(k,n),k=1..n) ;

%p end proc:

%t 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 *)

%Y Cf. A013661, A165430, A256392.

%K nonn

%O 1,2

%A _R. J. Mathar_, Jul 21 2016