A331131 - OEIS

%I #16 Mar 05 2020 03:39:54

%S 3,9,21,30,48,60,80,104,131,143,179,197,215,251,311,326,362,380,416,

%T 470,506,518,578,626,653,693,765,783,837,861,903,957,993,1029,1173,

%U 1200,1218,1254,1374,1398,1452,1476,1512,1620,1674,1686,1776,1836,1890,1944,2016,2034,2094,2166,2286,2358,2394

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

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

%H Amiram Eldar, <a href="/A331131/b331131.txt">Table of n, a(n) for n = 1..10000</a>

%H Richard Bellman, <a href="https://doi.org/10.1093/qmath/1.1.136">On some divisor sums associated with Diophantine equations</a>, The Quarterly Journal of Mathematics, Ser. 2, Vol. 1 (1950), pp. 136-146.

%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, Vol. 3, No. 1 (1957), pp. 396-413.

%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).

%F a(n) ~ c * n * log(n)^3, where c is a constant. - _Amiram Eldar_, Mar 05 2020

%t 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 *)

%Y Cf. A000005, A007425.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jan 11 2020