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

Author

Amiram Eldar, May 13 2022

Keywords

• nonn

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.
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A001622, A078473.

Programs

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]]

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).