OFFSET
1,2
COMMENTS
a(n) is the number of pairs of positive integers <= n with their LCM <= n. - Andrew Howroyd, Sep 01 2019
REFERENCES
Mentioned by Steven Finch in a posting to the Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Jun 13 2001.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1024
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, vol. 101, (2002), pp. 105-114.
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 19.
Vaclav Kotesovec, Graph - the asymptotic ratio
Eric Weisstein's World of Mathematics, Stieltjes Constants
FORMULA
a(n) = Sum_{j=1..n^2} floor(n/A019554(j)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002
a(n) = Sum_{i=1..n} 2^omega(i) * floor(n/i). - Enrique Pérez Herrero, Sep 15 2012
a(n) ~ 3/Pi^2 * n log^2 n. - Charles R Greathouse IV, Nov 08 2012
a(n) ~ 3*n/Pi^2 * (log(n)^2 + log(n)*(-2 + 6*g - 24*z/Pi^2) + 2 - 6*g + 6*g^2 - 6*sg1 + 288*z^2/Pi^4 - 24*(-z + 3*g*z + z2)/ Pi^2), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant (see A082633), z = Zeta'(2) (see A073002), z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Jan 30 2019
a(n) = Sum_{k=1..n} A064608(floor(n/k)). - Daniel Suteu, Mar 09 2019
MAPLE
with(numtheory): a:=n->add(tau(k^2), k=1..n): seq(a(n), n=1..60); # Muniru A Asiru, Mar 09 2019
MATHEMATICA
DivisorSigma[0, Range[60]^2] // Accumulate (* Jean-François Alcover, Nov 25 2013 *)
PROG
(PARI) for (n=1, 1024, write("b061503.txt", n, " ", sum(k=1, n, numdiv(k^2)))) \\ Harry J. Smith, Jul 23 2009
(PARI) t=0; v=vector(60, n, t+=numdiv(n^2)) \\ Charles R Greathouse IV, Nov 08 2012
(Sage) def A061503(n) :
tau = sloane.A000005
return add(tau(k^2) for k in (1..n))
[ A061503(i) for i in (1..19)] # Peter Luschny, Sep 15 2012
(GAP) List([1..60], n->Sum([1..n], k->Tau(k^2))); # Muniru A Asiru, Mar 09 2019
(Python)
from math import prod
from sympy import factorint
def A061503(n): return sum(prod(2*e+1 for e in factorint(k).values()) for k in range(1, n+1)) # Chai Wah Wu, May 10 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 14 2001
EXTENSIONS
Name corrected by Peter Luschny, Sep 15 2012
STATUS
approved