A268732 Sum of the numbers of divisors of gcd(x,y) with x*y <= n.

1, 3, 5, 9, 11, 15, 17, 23, 27, 31, 33, 41, 43, 47, 51, 60, 62, 70, 72, 80, 84, 88, 90, 102, 106, 110, 116, 124, 126, 134, 136, 148, 152, 156, 160, 176, 178, 182, 186, 198, 200, 208, 210, 218, 226, 230, 232, 250, 254, 262, 266, 274, 276, 288, 292, 304, 308, 312, 314, 330

Offset

1,2

Comments

Partial sums of A124315.

Links

Daniel Suteu, Table of n, a(n) for n = 1..10000

Masum Billal, Asymptotic Result of A Generalization of A GCD-Sum, arXiv:2206.05023 [math.NT], 2022.

Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, arXiv:1602.03555 [math.HO], 2016. See Remark 1 p. 3.

Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, The American Mathematical Monthly, Vol. 123, No. 9 (2016), 924-927.

Randell Heyman, A summation involving the number of divisors function and the GCD function, arXiv:2003.13937 [math.NT], 2020.

Formula

a(n) = Sum_{k=1..floor(sqrt(n))} (2*Sum_{j=1..floor(sqrt(n/k^2))} floor(n/(j*k^2)) - floor(sqrt(n/k^2))^2). - Daniel Suteu, Jan 08 2019

a(n) = n*zeta(2)*(log(n) + 2*gamma - 1 + 2*zeta'(2)/zeta(2)) + O(sqrt(n)*log(n)), where gamma is the Euler-Mascheroni constant A001620. - Daniel Suteu, Jan 11 2019

Mathematica

Table[Total@ Flatten@ Map[Function[k, DivisorSigma[0, GCD[#, k]] & /@ Select[Range@ n, # k <= n &]], Range@ n], {n, 60}] (* Michael De Vlieger, Feb 12 2016 *)

Prog

(PARI) a(n) = sum(k=1, n, sumdiv(k, d, numdiv(gcd(d, k/d))));
(PARI) a(n) = sum(k=1, sqrtint(n), 2*sum(j=1, sqrtint(n\(k*k)), n\(j*k*k))-sqrtint(n\(k*k))^2); \\ Daniel Suteu, Jan 08 2019
(PARI) a(n)=sum(k=1, n, sum(j=1, sqrt(n/k), floor(n/k/j^2))); \\ Benoit Cloitre, Oct 02 2022

Crossrefs

Cf. A000005, A124315.

Cf. A001620, A013661, A306016.

Sequence in context: A164121 A333171 A078651 * A101114 A120696 A071156

Adjacent sequences: A268729 A268730 A268731 * A268733 A268734 A268735

Keyword

nonn

Author

Michel Marcus, Feb 12 2016

Status

approved