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

Author

Amiram Eldar, Sep 06 2024

Keywords

• nonn
• base

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.

Examples

			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.
		

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.

Crossrefs

Cf. A352089, A352090.

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

Programs

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]