A078473 -id:A078473 - cp's OEIS frontend

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

Amiram Eldar, May 13 2022

nonn

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)

Cf. A001622, A078473.

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]]

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).

A035282 Expansion of zeta function of icosian ring (nonzero terms).

Original entry on oeis.org

1, 5, 6, 10, 24, 21, 40, 30, 31, 60, 64, 50, 84, 120, 60, 50, 144, 120, 124, 85, 144, 200, 160, 126, 91, 180, 240, 240, 155, 204, 220, 300, 410, 320, 156, 264, 280, 210, 360, 300, 304, 384, 420, 170, 400, 504, 360, 300, 364, 384, 250, 400, 504, 960, 424, 720, 310

Offset: 1

N. J. A. Sloane

Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187; then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

Nonzero terms of A078473. - Michel Marcus, Mar 03 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. (1999), 51 1258-1276.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.

Cf. A031363, A035187, A078472, A078473.

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]; Select[Array[s, 200], # > 0 &] (* Amiram Eldar, May 13 2022 *)

A035111 Numerators in expansion of a certain Dirichlet series.

Original entry on oeis.org

1, 5, 6, 10, 24, 20, 40, 30, 30, 60, 64, 50, 84, 120, 60, 50, 144, 120, 124, 80, 144, 200, 160, 120, 90, 180, 240, 240, 150, 204, 220, 300, 408, 320, 150, 264, 280, 200, 360, 300, 304, 384, 420, 170, 400, 480, 360, 300, 364, 384, 250, 400, 504, 960, 424, 720, 300

Offset: 0

N. J. A. Sloane

This is the SO(3) case, whereas the SO(4) case is reported in A078473 and A035282. [From R. J. Mathar, Jul 16 2010]

M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.

More terms from R. J. Mathar, Jul 16 2010

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A353997 Partial sums of A078473.

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A035282 Expansion of zeta function of icosian ring (nonzero terms).

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A035111 Numerators in expansion of a certain Dirichlet series.

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