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A284372 a(n) = Sum_{d|n, d = 0, 1, or 11 mod 12} d.

%I A284372 #23 Aug 29 2024 17:06:01
%S A284372 1,1,1,1,1,1,1,1,1,1,12,13,14,1,1,1,1,1,1,1,1,12,24,37,26,14,1,1,1,1,
%T A284372 1,1,12,1,36,49,38,1,14,1,1,1,1,12,1,24,48,85,50,26,1,14,1,1,12,1,1,1,
%U A284372 60,73,62,1,1,1,14,12,1,1,24,36,72,145,74,38,26,1
%N A284372 a(n) = Sum_{d|n, d = 0, 1, or 11 mod 12} d.
%H A284372 Seiichi Manyama, <a href="/A284372/b284372.txt">Table of n, a(n) for n = 1..10000</a>
%F A284372 From _Peter Bala_, Dec 11 2020: (Start)
%F A284372 O.g.f.: Sum_{k >= 1, k == 0, 1 or 11 (mod 12)} k*x^k/(1 - x^k).
%F A284372 Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - -, where [1, 11, 14, 34, ...] is the sequence of generalized 14-gonal numbers A195818, and e(n) = (-1)^(m+1)*n if n is a generalized 14-gonal number of the form m*(6*m+-5); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
%F A284372 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/48 = -A245058 = 0.205616... . - _Amiram Eldar_, Apr 12 2024
%e A284372 From _Peter Bala_, Dec 11 2020: (Start)
%e A284372 n = 24: n is not of the form m*(6*m +- 5), so e(n) = 0 and a(24) = a(23) + a(13) - a(10)  = 24 + 14 - 1  = 37;
%e A284372 n = 39: n = m*(6*m - 5) for m = 3, so e(n) = 39 and a(39) = 39 + a(38) + a(28) - a(25) - a(5) = 39 + 1 + 1 - 26 - 1 = 14;
%e A284372 n = 76: n = m*(6*m - 5) for m = 4, so e(n) = -76 and a(4) = -76 + a(75) + a(65) - a(62) - a(42) + a(37) + a(7) = -76 + 26 + 14  - 1 - 1 + 38 + 1 = 1. (End)
%t A284372 Table[Sum[If[Mod[d, 12]<2 || Mod[d, 12]==11, d, 0], {d, Divisors[n]}], {n, 80}] (* _Indranil Ghosh_, Mar 25 2017 *)
%t A284372 sd12[n_]:=Total[Select[Divisors[n],MemberQ[{0,1,11},Mod[#,12]]&]]; Array[sd12,80] (* _Harvey P. Dale_, Aug 29 2024 *)
%o A284372 (PARI) a(n) = sumdiv(n, d, ((d + 1) % 12 < 3) * d); \\ _Amiram Eldar_, Apr 12 2024
%Y A284372 Cf. A210964 (1/f(-x, -x^11)), A245058.
%Y A284372 Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A113184(k=4), A284361 (k=5), A284362 (k=6), A284363 (k=7), this sequence (k=12).
%K A284372 nonn,easy
%O A284372 1,11
%A A284372 _Seiichi Manyama_, Mar 25 2017