A331133 a(n) = Sum_{i=1..n} d(i)^2*d(i+1)/2, where d(n) = A000005(n).

1, 5, 11, 20, 28, 44, 52, 76, 94, 110, 122, 158, 166, 198, 238, 263, 275, 311, 323, 395, 427, 443, 459, 555, 573, 605, 653, 689, 705, 769, 781, 853, 885, 917, 989, 1070, 1078, 1110, 1174, 1238, 1254, 1318, 1330, 1438, 1510, 1526, 1546, 1696, 1723, 1795, 1843, 1879, 1895, 2023, 2087, 2215, 2247, 2263, 2287, 2431

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Yoichi Motohashi, An asymptotic formula in the theory of numbers, Acta Arith. 16 (1969/70), 255-264. MR0266884 (42 #1786).

Formula

From Amiram Eldar, Apr 23 2024: (Start)

a(n) = A331132(n)/2.

a(n) ~ (c/(2*Pi^2)) * n * log(n)^4, where c = Product_{p prime} (1 - 1/p + (1/p)*(1 - 1/p)^2*(1 + 1/p)^(-1)) = 0.35823163000196141456... . (End)

Mathematica

Accumulate[a[n_]:=DivisorSum[n+1, DivisorSigma[0, n]^2 &]/2; Array[a, 60]] (* Vincenzo Librandi, Jan 12 2020 *)

Prog

(Magma) [&+[#Divisors(i)^2*#Divisors(i+1)/2:i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Jan 12 2020
(PARI) lista(nmax) = {my(d1 = 1, d2, s = 0); for(n = 2, nmax, d2 = numdiv(n); s += d1^2 * d2; print1(s/2, ", "); d1 = d2); } \\ Amiram Eldar, Apr 23 2024

Crossrefs

Cf. A000005, A331132.

Sequence in context: A029456 A304502 A190743 * A110208 A034308 A032527

Adjacent sequences: A331130 A331131 A331132 * A331134 A331135 A331136

Keyword

nonn

Author

N. J. A. Sloane, Jan 11 2020

Status

approved