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

%I A061503 #61 Feb 16 2025 08:32:44
%S A061503 1,4,7,12,15,24,27,34,39,48,51,66,69,78,87,96,99,114,117,132,141,150,
%T A061503 153,174,179,188,195,210,213,240,243,254,263,272,281,306,309,318,327,
%U A061503 348,351,378,381,396,411,420,423,450,455,470,479,494,497
%N A061503 a(n) = Sum_{k=1..n} tau(k^2), where tau is the number of divisors function A000005.
%C A061503 a(n) is the number of pairs of positive integers <= n with their LCM <= n. - _Andrew Howroyd_, Sep 01 2019
%D A061503 Mentioned by _Steven Finch_ in a posting to the Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Jun 13 2001.
%H A061503 Harry J. Smith, <a href="/A061503/b061503.txt">Table of n, a(n) for n = 1..1024</a>
%H A061503 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 A061503 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 19.
%H A061503 Vaclav Kotesovec, <a href="/A061503/a061503.jpg">Graph - the asymptotic ratio</a>
%H A061503 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>
%F A061503 a(n) = Sum_{j=1..n^2} floor(n/A019554(j)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002
%F A061503 a(n) = Sum_{i=1..n} 2^omega(i) * floor(n/i). - _Enrique Pérez Herrero_, Sep 15 2012
%F A061503 a(n) ~ 3/Pi^2 * n log^2 n. - _Charles R Greathouse IV_, Nov 08 2012
%F A061503 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 A061503 a(n) = Sum_{k=1..n} A064608(floor(n/k)). - _Daniel Suteu_, Mar 09 2019
%p A061503 with(numtheory): a:=n->add(tau(k^2),k=1..n): seq(a(n),n=1..60); # _Muniru A Asiru_, Mar 09 2019
%t A061503 DivisorSigma[0, Range[60]^2] // Accumulate (* _Jean-François Alcover_, Nov 25 2013 *)
%o A061503 (PARI) for (n=1, 1024, write("b061503.txt", n, " ", sum(k=1, n, numdiv(k^2)))) \\ _Harry J. Smith_, Jul 23 2009
%o A061503 (PARI) t=0;v=vector(60,n,t+=numdiv(n^2)) \\ _Charles R Greathouse IV_, Nov 08 2012
%o A061503 (Sage) def A061503(n) :
%o A061503     tau = sloane.A000005
%o A061503     return add(tau(k^2) for k in (1..n))
%o A061503 [ A061503(i) for i in (1..19)] # _Peter Luschny_, Sep 15 2012
%o A061503 (GAP) List([1..60],n->Sum([1..n],k->Tau(k^2))); # _Muniru A Asiru_, Mar 09 2019
%o A061503 (Python)
%o A061503 from math import prod
%o A061503 from sympy import factorint
%o A061503 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 A061503 Cf. A000005, A061502. Partial sums of A048691.
%K A061503 nonn
%O A061503 1,2
%A A061503 _N. J. A. Sloane_, Jun 14 2001
%E A061503 Name corrected by _Peter Luschny_, Sep 15 2012