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