A353997 Partial sums of A078473.

1, 1, 1, 6, 12, 12, 12, 12, 22, 22, 46, 46, 46, 46, 46, 67, 67, 67, 107, 137, 137, 137, 137, 137, 168, 168, 168, 168, 228, 228, 292, 292, 292, 292, 292, 342, 342, 342, 342, 342, 426, 426, 426, 546, 606, 606, 606, 606, 656, 656, 656, 656, 656, 656, 800, 800, 800

Offset

1,4

Links

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

Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math., Vol. 51, No. 6 (1999), pp. 1258-1276; arXiv preprint, arXiv:math/9904028 [math.MG], 1999.

Formula

a(n) = Sum_{k=1..n} A078473(k).

a(n) ~ c*n^2 where c = 2*Pi^4*log(phi)/375 = 0.2499968345... and phi is the golden ratio (1+sqrt(5))/2 (A001622) (Baake and Moody, 1999).

Mathematica

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]]

Crossrefs

Cf. A001622, A078473.

Sequence in context: A315568 A315569 A315570 * A315571 A135462 A315572

Adjacent sequences: A353994 A353995 A353996 * A353998 A353999 A354000

Keyword

nonn

Author

Amiram Eldar, May 13 2022

Status

approved