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%I #8 May 14 2022 07:44:25
%S 1,1,1,6,12,12,12,12,22,22,46,46,46,46,46,67,67,67,107,137,137,137,
%T 137,137,168,168,168,168,228,228,292,292,292,292,292,342,342,342,342,
%U 342,426,426,426,546,606,606,606,606,656,656,656,656,656,656,800,800,800
%N Partial sums of A078473.
%H Amiram Eldar, <a href="/A353997/b353997.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Baake and Robert V. Moody, <a href="http://dx.doi.org/10.4153/CJM-1999-057-0">Similarity submodules and root systems in four dimensions</a>, Canad. J. Math., Vol. 51, No. 6 (1999), pp. 1258-1276; <a href="https://arxiv.org/abs/math/9904028">arXiv preprint</a>, arXiv:math/9904028 [math.MG], 1999.
%F a(n) = Sum_{k=1..n} A078473(k).
%F 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).
%t 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]]
%Y Cf. A001622, A078473.
%K nonn
%O 1,4
%A _Amiram Eldar_, May 13 2022
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