A376029 - OEIS

%I #11 Sep 10 2024 08:44:04

%S 1,2,8,48,140,244,620,705,1395,6210,9656,14322,52024,88128,95589,

%T 119440,151130,325105,407128,472790,520971,686103,4456608,7066416,

%U 13207389,15488160,23381160,42317212,49496700,53564016,128163495,165750096,387900480,421730960,485880806

%N Tribonacci-Niven numbers (A352089) with a record gap to the next tribonacci-Niven number.

%C The corresponding record gaps are 1, 2, 4, 8, 9, 12, 15, 20, ... (see the link for more values).

%C Ray (2005) and Ray and Cooper (2006) proved that the asymptotic density of the tribonacci-Niven numbers is 0. Therefore, this sequence is infinite.

%D Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.

%H Amiram Eldar, <a href="/A376029/b376029.txt">Table of n, a(n) for n = 1..42</a>

%H Amiram Eldar, <a href="/A376029/a376029.txt">Table of n, a(n), gap(n) for n = 1..42</a>

%H Andrew Ray and Curtis Cooper, <a href="http://cs.ucmo.edu/~cnc8851/articles/kzeckniven.pdf">On the natural density of the k-Zeckendorf Niven numbers</a>, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.

%e The first 7 tribonacci-Niven numbers are 1, 2, 4, 6, 7, 8 and 12. The gaps between them are 1, 2, 2, 1, 1 and 4, and the record gaps, 1, 2 and 4 occur after 1, 2 and 8, the first 3 terms of this sequence.

%t t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; tnQ[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; s++; m -= t[k]; k = 1]; Divisible[n, s]]; seq[kmax_] := Module[{gapmax = 0, gap, k1 = 1, s = {}}, Do[If[tnQ[k], gap = k - k1; If[gap > gapmax, gapmax = gap; AppendTo[s, k1]]; k1 = k], {k, 2, kmax}]; s]; seq[10^4]

%Y Cf. A352089, A352090.

%Y Similar sequences: A337076, A337077, A347495, A347496, A376028.

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Sep 06 2024