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

1,2

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

Cf. A000005, A061502. Partial sums of A048691.

Sequence in context: A310778 A345428 A350244 * A134659 A075624 A008333

Adjacent sequences: A061500 A061501 A061502 * A061504 A061505 A061506

Keyword

nonn

Author

N. J. A. Sloane, Jun 14 2001

Extensions

Name corrected by Peter Luschny, Sep 15 2012

Status

approved