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%I #22 Apr 19 2024 03:29:14
%S 1,10,19,55,64,145,154,254,290,371,380,704,713,794,875,1100,1109,1433,
%T 1442,1766,1847,1928,1937,2837,2873,2954,3054,3378,3387,4116,4125,
%U 4566,4647,4728,4809,6105,6114,6195,6276,7176,7185,7914,7923,8247,8571,8652,8661,10686,10722,11046,11127,11451,11460,12360
%N Partial sums of A097988 (d_3(n)^2).
%C This and the following sequences (and continuing in A331071) were inspired by the papers of Hooley, Indlekofer, Motohashi, Redmond, Titchmarsh, etc.
%H Seiichi Manyama, <a href="/A330570/b330570.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from Vincenzo Librandi)
%H V. C. Harris and M. V. Subbarao, <a href="https://doi.org/10.1216/RMJ-1985-15-2-399">On the divisor sum function</a>, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp 399-412; <a href="https://www.jstor.org/stable/44236923">alternative link</a>.
%H C. Hooley, <a href="https://doi.org/10.1112/plms/s3-7.1.396">An Asymptotic Formula in the Theory of Numbers</a>, Proceedings of the London Mathematical Society, Volume s3-7, Issue 1, 1957, Pages 396-413.
%H Karl-Heinz Indlekofer, <a href="https://doi.org/10.1007/BF01304942">Eine asymptotische Formel in der Zahlentheorie</a> (German), Arch. Math. (Basel) 23 (1972), 619-624. MR0318080 (47 #6629).
%H Yoichi Motohashi, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1633.pdf">An asymptotic formula in the theory of numbers</a>, Acta Arith. 16 (1969/70), 255-264. MR0266884 (42 #1786).
%H Don Redmond, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002313499">An asymptotic formula in the theory of numbers</a>, Math. Ann. 224 (1976), no. 3, 247-268. MR0419386 (54 #7407).
%H E. C. Titchmarsh, <a href="https://doi.org/10.1093/qmath/os-13.1.129">Some problems in the analytic theory of numbers</a>, The Quarterly Journal of Mathematics 1 (1942): 129-152.
%F a(n) ~ c * n * log(n)^8 /8!, where c = Product_{p prime} ((1-1/p)^4 * (1 + 4/p + 1/p^2)) = 0.049321673579400091761... (Titchmarsh, 1942). - _Amiram Eldar_, Apr 19 2024
%t Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, #]&]^2; Array[a, 60]] (* _Vincenzo Librandi_, Jan 11 2020 *)
%o (PARI) lista(nmax) = {my(s = 0); for(n = 1, nmax, s += vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[,2]))^2; print1(s, ", "));} \\ _Amiram Eldar_, Apr 19 2024
%Y Cf. A007425, A097988.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jan 08 2020
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