A330320 a(n) = Sum_{i=1..n} tau(i)*tau(i+1), where tau(n) = A000005(n) is the number of divisors of n.

2, 6, 12, 18, 26, 34, 42, 54, 66, 74, 86, 98, 106, 122, 142, 152, 164, 176, 188, 212, 228, 236, 252, 276, 288, 304, 328, 340, 356, 372, 384, 408, 424, 440, 476, 494, 502, 518, 550, 566, 582, 598, 610, 646, 670, 678, 698, 728, 746, 770, 794, 806, 822, 854, 886, 918, 934, 942, 966, 990, 998, 1022, 1064, 1092, 1124, 1140

Offset: 1

Author

N. J. A. Sloane, Dec 11 2019

Keywords

• nonn

Comments

For background references see A330570.

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 61.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. 1, No. 3 (1927), pp. 202-208.
• Nikolay V. Kuznetsov, Convolution of the Fourier coefficients of the Eisenstein-Maass series, Journal of Soviet Mathematics, Vol. 29, No. 2 (1985), pp. 1131-1159.

Crossrefs

Cf. A000005, A059956, A063123.

Partial sums of A092517.

Programs

Mathematica

Accumulate[a[n_]:=DivisorSum[n+1, DivisorSigma[0, n]&]; Array[a, 66]] (* Vincenzo Librandi, Jan 10 2020 *)
Accumulate[Times@@@Partition[DivisorSigma[0,Range[70]],2,1]] (* Harvey P. Dale, Nov 02 2023 *)

PARI

a(n) = sum(i=1, n, numdiv(i*(i+1))); \\ Michel Marcus, Jan 11 2020

Formula

a(n) ~ (1/zeta(2)) * n * log(n)^2. - Amiram Eldar, Mar 05 2020