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
KEYWORD
nonn
AUTHOR
Michel Marcus, Feb 12 2016
STATUS
approved