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a(n) = Sum_{k=1..n} tau(k^2), where tau is the number of divisors function A000005.
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%I #60 May 10 2022 17:05:33

%S 1,4,7,12,15,24,27,34,39,48,51,66,69,78,87,96,99,114,117,132,141,150,

%T 153,174,179,188,195,210,213,240,243,254,263,272,281,306,309,318,327,

%U 348,351,378,381,396,411,420,423,450,455,470,479,494,497

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

%C a(n) is the number of pairs of positive integers <= n with their LCM <= n. - _Andrew Howroyd_, Sep 01 2019

%D Mentioned by _Steven Finch_ in a posting to the Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Jun 13 2001.

%H Harry J. Smith, <a href="/A061503/b061503.txt">Table of n, a(n) for n = 1..1024</a>

%H Kevin A. Broughan, <a href="http://www.math.waikato.ac.nz/~kab/papers/div4.pdf">Restricted divisor sums</a>, Acta Arithmetica, vol. 101, (2002), pp. 105-114.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 19.

%H Vaclav Kotesovec, <a href="/A061503/a061503.jpg">Graph - the asymptotic ratio</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>

%F a(n) = Sum_{j=1..n^2} floor(n/A019554(j)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002

%F a(n) = Sum_{i=1..n} 2^omega(i) * floor(n/i). - _Enrique Pérez Herrero_, Sep 15 2012

%F a(n) ~ 3/Pi^2 * n log^2 n. - _Charles R Greathouse IV_, Nov 08 2012

%F 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

%F a(n) = Sum_{k=1..n} A064608(floor(n/k)). - _Daniel Suteu_, Mar 09 2019

%p with(numtheory): a:=n->add(tau(k^2),k=1..n): seq(a(n),n=1..60); # _Muniru A Asiru_, Mar 09 2019

%t DivisorSigma[0, Range[60]^2] // Accumulate (* _Jean-François Alcover_, Nov 25 2013 *)

%o (PARI) for (n=1, 1024, write("b061503.txt", n, " ", sum(k=1, n, numdiv(k^2)))) \\ _Harry J. Smith_, Jul 23 2009

%o (PARI) t=0;v=vector(60,n,t+=numdiv(n^2)) \\ _Charles R Greathouse IV_, Nov 08 2012

%o (Sage) def A061503(n) :

%o tau = sloane.A000005

%o return add(tau(k^2) for k in (1..n))

%o [ A061503(i) for i in (1..19)] # _Peter Luschny_, Sep 15 2012

%o (GAP) List([1..60],n->Sum([1..n],k->Tau(k^2))); # _Muniru A Asiru_, Mar 09 2019

%o (Python)

%o from math import prod

%o from sympy import factorint

%o 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

%Y Cf. A000005, A061502. Partial sums of A048691.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jun 14 2001

%E Name corrected by _Peter Luschny_, Sep 15 2012