A376029 Tribonacci-Niven numbers (A352089) with a record gap to the next tribonacci-Niven number.

1, 2, 8, 48, 140, 244, 620, 705, 1395, 6210, 9656, 14322, 52024, 88128, 95589, 119440, 151130, 325105, 407128, 472790, 520971, 686103, 4456608, 7066416, 13207389, 15488160, 23381160, 42317212, 49496700, 53564016, 128163495, 165750096, 387900480, 421730960, 485880806

Offset

1,2

Comments

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

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

References

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..42

Amiram Eldar, Table of n, a(n), gap(n) for n = 1..42

Andrew Ray and Curtis Cooper, On the natural density of the k-Zeckendorf Niven numbers, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.

Example

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.

Mathematica

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]

Crossrefs

Cf. A352089, A352090.

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

Sequence in context: A003032 A193944 A058928 * A228288 A356346 A292277

Adjacent sequences: A376026 A376027 A376028 * A376030 A376031 A376032

Keyword

nonn,base

Author

Amiram Eldar, Sep 06 2024

Status

approved