A331077 a(n) = Sum_{k = 1..n} [d(k)*d_3(k)], where d = A000005, d_3 = A007425.

1, 7, 13, 31, 37, 73, 79, 119, 137, 173, 179, 287, 293, 329, 365, 440, 446, 554, 560, 668, 704, 740, 746, 986, 1004, 1040, 1080, 1188, 1194, 1410, 1416, 1542, 1578, 1614, 1650, 1974, 1980, 2016, 2052, 2292, 2298, 2514, 2520, 2628, 2736, 2772, 2778, 3228, 3246, 3354, 3390, 3498, 3504, 3744, 3780, 4020, 4056

Offset

1,2

Comments

For background references see A330570.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Formula

a(n) ~ c * n * log(n)^5 /5!, where c = Product_{p prime} ((1-1/p)^2*(1+2/p)) = 0.286747428434478734107... (Titchmarsh, 1942). - Amiram Eldar, Apr 19 2024

Mathematica

f[p_, e_] := (e+1)^2*(e+2)/2; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* Amiram Eldar, Apr 19 2024 *)

Prog

(PARI) lista(nmax) = {my(s = 0); for(n = 1, nmax, s += vecprod(apply(e -> (e+1)^2*(e+2)/2, factor(n)[, 2])); print1(s, ", ")); } \\ Amiram Eldar, Apr 19 2024

Crossrefs

Cf. A000005, A007425.

Sequence in context: A046139 A023243 A335794 * A087196 A074963 A283937

Adjacent sequences: A331074 A331075 A331076 * A331078 A331079 A331080

Keyword

nonn

Author

N. J. A. Sloane, Jan 10 2020

Status

approved