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A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.

%I A061502 #65 Sep 08 2022 08:45:03
%S A061502 1,5,9,18,22,38,42,58,67,83,87,123,127,143,159,184,188,224,228,264,
%T A061502 280,296,300,364,373,389,405,441,445,509,513,549,565,581,597,678,682,
%U A061502 698,714,778,782,846,850,886,922,938,942,1042,1051,1087
%N A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.
%D A061502 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 56.
%H A061502 N. J. A. Sloane, <a href="/A061502/b061502.txt">Table of n, a(n) for n = 1..1024</a>
%H A061502 Adrian Dudek, <a href="http://arxiv.org/abs/1602.03555">On the Success of Mishandling Euclid's Lemma</a>, arXiv:1602.03555 [math.HO], 2016. See B(n) p. 2.
%H A061502 Chaohua Jia and Ayyadurai Sankaranarayanan, <a href="http://doi.org/10.4064/aa164-2-7">The mean square of the divisor function</a>, Acta Arithmetica 164 (2014), 181-208.
%H A061502 Michaela Cully-Hugill and Timothy Trudgian, <a href="https://arxiv.org/abs/1911.07369">Two explicit divisor sums</a>, arXiv:1911.07369 [math.NT], Nov 19 2019
%H A061502 Vaclav Kotesovec, <a href="/A061502/a061502_1.jpg">Graph - The asymptotic ratio (1000000 terms)</a>
%H A061502 Florian Luca and László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Luca/luca42.html">The r-th moment of the divisor function: an elementary approach</a>, Journal of Integer Sequences 20 (2017), Article 17.7.4, 8 pp.
%H A061502 Adolf Piltz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN271032898">Über das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst</a>, 1881.
%H A061502 Ramanujan's Papers, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulas in the analytic theory of numbers</a>, Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).
%H A061502 D. Suryanarayana and R. Rama Chandra Rao, <a href="https://doi.org/10.7146/math.scand.a-11460">On an Asymptotic Formula of Ramanujan</a>, Mathematica Scandinavica, 32, 258-264, 1973.
%H A061502 B. M. Wilson, <a href="http://plms.oxfordjournals.org/content/s2-21/1/235.extract">Proofs of some formulas enunciated by Ramanujan</a>, Proc. London Math. Soc. (2) 21 (1922) 235-255.
%F A061502 a(n) = Sum_{k=1..n} tau(k^2)*floor(n/k).
%F A061502 Asymptotic to A*n*log(n)^3 + B*n*log(n)^2 + C*n*log(n) + D*n + O(n^(1/2+eps)) where A = 1/Pi^2 and B = (12*gamma-3)/Pi^2 - 36*zeta'(2)/Pi^4. [corrected by _Vaclav Kotesovec_, Aug 30 2018]
%F A061502 C = 36*gamma^2/Pi^2 - (288*z1/Pi^4 + 24/Pi^2)*gamma + (864*z1^2/Pi^6 + 72*z1/Pi^4 - 72/Pi^4*z2 + 6/Pi^2) - 24*g1/Pi^2 and D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279. - _Vaclav Kotesovec_, Sep 10 2018
%F A061502 See Cully-Hugill & Trudgian, Theorem 2, for an explicit version of the asymptotic given above. - _Charles R Greathouse IV_, Nov 19 2019
%t A061502 Table[Sum[DivisorSigma[0, k^2]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* _Vaclav Kotesovec_, Aug 30 2018 *)
%t A061502 Accumulate[Table[DivisorSigma[0, n]^2, {n, 1, 50}]] (* _Vaclav Kotesovec_, Sep 10 2018 *)
%o A061502 (PARI) for (n=1, 1024, write("b061502.txt", n, " ", sum(k=1, n, numdiv(k)^2)) ) \\ _Harry J. Smith_, Jul 23 2009
%o A061502 (PARI) vector(60, n, sum(k=1, n, numdiv(k)^2)) \\ _Michel Marcus_, Jul 23 2015
%o A061502 (PARI) first(n)=my(v=vector(n),s); forfactored(k=1,n, v[k[1]] = s += numdiv(k)^2); v; \\ _Charles R Greathouse IV_, Nov 28 2018
%o A061502 (Magma) [&+[NumberOfDivisors(k^2)*Floor(n/k): k in [1..n]]: n in [1..60]]; // _Vincenzo Librandi_, Sep 10 2016
%Y A061502 Cf. A000005, A035116, A061503, A318755.
%Y A061502 Cf. A092742 (A), A245074 (B), A319090 (C), A319091 (D).
%Y A061502 Cf. A057434, A072379, A074789.
%K A061502 nonn
%O A061502 1,2
%A A061502 _N. J. A. Sloane_, Jun 14 2001
%E A061502 Definition corrected by _N. J. A. Sloane_, May 25 2008