A331131 a(n) = Sum_{i=1..n} d(i)*d_3(i+1), where d(n) = A000005(n) and d_3(n) = A007425(n).

3, 9, 21, 30, 48, 60, 80, 104, 131, 143, 179, 197, 215, 251, 311, 326, 362, 380, 416, 470, 506, 518, 578, 626, 653, 693, 765, 783, 837, 861, 903, 957, 993, 1029, 1173, 1200, 1218, 1254, 1374, 1398, 1452, 1476, 1512, 1620, 1674, 1686, 1776, 1836, 1890, 1944, 2016, 2034, 2094, 2166, 2286, 2358, 2394

Offset: 1

Author

N. J. A. Sloane, Jan 11 2020

Keywords

• nonn

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 62.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Richard Bellman, On some divisor sums associated with Diophantine equations, The Quarterly Journal of Mathematics, Ser. 2, Vol. 1 (1950), pp. 136-146.
• C. Hooley, An asymptotic formula in the theory of numbers, Proceedings of the London Mathematical Society, Vol. 3, No. 1 (1957), pp. 396-413.
• Yoichi Motohashi, An asymptotic formula in the theory of numbers, Acta Arith. 16 (1969/70), 255-264. MR0266884 (42 #1786).

Crossrefs

Cf. A000005, A007425.

Programs

Mathematica

f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate @ Table[DivisorSigma[0, n] * d3[n+1], {n, 1, 60}] (* Amiram Eldar, Mar 05 2020 *)

Formula

a(n) ~ c * n * log(n)^3, where c is a constant. - Amiram Eldar, Mar 05 2020