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
Keywords
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.
Programs
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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 *) -
PARI
a(n) = sum(k=1, n, sumdiv(k, d, numdiv(gcd(d, k/d))));
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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
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PARI
a(n)=sum(k=1,n,sum(j=1,sqrt(n/k),floor(n/k/j^2))); \\ Benoit Cloitre, Oct 02 2022
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
a(n) = Sum_{i=1..n} Sum_{j=1..n} floor(sqrt(n/(i*j))). - Ridouane Oudra, Apr 13 2025
Comments