A061503 a(n) = Sum_{k=1..n} tau(k^2), where tau is the number of divisors function A000005.

1, 4, 7, 12, 15, 24, 27, 34, 39, 48, 51, 66, 69, 78, 87, 96, 99, 114, 117, 132, 141, 150, 153, 174, 179, 188, 195, 210, 213, 240, 243, 254, 263, 272, 281, 306, 309, 318, 327, 348, 351, 378, 381, 396, 411, 420, 423, 450, 455, 470, 479, 494, 497

Offset: 1

Author

N. J. A. Sloane, Jun 14 2001

Keywords

• nonn

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

Crossrefs

Cf. A000005, A061502. Partial sums of A048691.

Programs

GAP

List([1..60],n->Sum([1..n],k->Tau(k^2))); # Muniru A Asiru, 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 *)

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

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

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

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

Extensions

Name corrected by Peter Luschny, Sep 15 2012