A330570 Partial sums of A097988 (d_3(n)^2).

1, 10, 19, 55, 64, 145, 154, 254, 290, 371, 380, 704, 713, 794, 875, 1100, 1109, 1433, 1442, 1766, 1847, 1928, 1937, 2837, 2873, 2954, 3054, 3378, 3387, 4116, 4125, 4566, 4647, 4728, 4809, 6105, 6114, 6195, 6276, 7176, 7185, 7914, 7923, 8247, 8571, 8652, 8661, 10686, 10722, 11046, 11127, 11451, 11460, 12360

Offset

1,2

Comments

This and the following sequences (and continuing in A331071) were inspired by the papers of Hooley, Indlekofer, Motohashi, Redmond, Titchmarsh, etc.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Vincenzo Librandi)

V. C. Harris and M. V. Subbarao, On the divisor sum function, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp 399-412; alternative link.

C. Hooley, An Asymptotic Formula in the Theory of Numbers, Proceedings of the London Mathematical Society, Volume s3-7, Issue 1, 1957, Pages 396-413.

Karl-Heinz Indlekofer, Eine asymptotische Formel in der Zahlentheorie (German), Arch. Math. (Basel) 23 (1972), 619-624. MR0318080 (47 #6629).

Yoichi Motohashi, An asymptotic formula in the theory of numbers, Acta Arith. 16 (1969/70), 255-264. MR0266884 (42 #1786).

Don Redmond, An asymptotic formula in the theory of numbers, Math. Ann. 224 (1976), no. 3, 247-268. MR0419386 (54 #7407).

E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.

Formula

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

Mathematica

Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, #]&]^2; Array[a, 60]] (* Vincenzo Librandi, Jan 11 2020 *)

Prog

(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

Crossrefs

Cf. A007425, A097988.

Sequence in context: A088409 A219959 A307344 * A369305 A065198 A033866

Adjacent sequences: A330567 A330568 A330569 * A330571 A330572 A330573

Keyword

nonn,changed

Author

N. J. A. Sloane, Jan 08 2020

Status

approved