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

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, 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, 60, 73, 62, 1, 1, 1, 14, 12, 1, 1, 24, 36, 72, 145, 74, 38, 26, 1

Offset

1,11

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

From Peter Bala, Dec 11 2020: (Start)

O.g.f.: Sum_{k >= 1, k == 0, 1 or 11 (mod 12)} k*x^k/(1 - x^k).

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)

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/48 = -A245058 = 0.205616... . - Amiram Eldar, Apr 12 2024

Example

From Peter Bala, Dec 11 2020: (Start)
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;
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;
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)

Mathematica

Table[Sum[If[Mod[d, 12]<2 || Mod[d, 12]==11, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
sd12[n_]:=Total[Select[Divisors[n], MemberQ[{0, 1, 11}, Mod[#, 12]]&]]; Array[sd12, 80] (* Harvey P. Dale, Aug 29 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, ((d + 1) % 12 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Crossrefs

Cf. A210964 (1/f(-x, -x^11)), A245058.

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).

Sequence in context: A105745 A020511 A008939 * A004453 A004501 A071955

Adjacent sequences: A284369 A284370 A284371 * A284373 A284374 A284375

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 25 2017

Status

approved