A352089 Tribonacci-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the tribonacci numbers (A278038).

1, 2, 4, 6, 7, 8, 12, 13, 14, 18, 20, 21, 24, 26, 27, 28, 30, 33, 36, 39, 40, 44, 46, 48, 56, 60, 68, 69, 72, 75, 76, 80, 81, 82, 84, 87, 88, 90, 94, 96, 100, 108, 115, 116, 120, 126, 128, 129, 132, 135, 136, 138, 140, 149, 150, 156, 162, 168, 174, 176, 177, 180

Offset

1,2

Comments

Numbers k such that A278043(k) | k.

The positive tribonacci numbers (A000073) are all terms.

If k = A000073(A042964(m)) is an odd tribonacci number, then k+1 is a term.

Ray (2005) and Ray and Cooper (2006) called these numbers "3-Zeckendorf Niven numbers" and proved that their asymptotic density is 0. - Amiram Eldar, Sep 06 2024

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..10000

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

6 is a term since its minimal tribonacci representation, A278038(6) = 110, has A278043(6) = 2 1's and 6 is divisible by 2.

Mathematica

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; Divisible[n, DigitCount[Total[2^(s - 1)], 2, 1]]]; Select[Range[180], q]

Crossrefs

Cf. A000073, A042964, A278038, A278043.

Subsequences: A352090, A352091, A352092.

Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719.

Sequence in context: A186112 A029453 A330943 * A358458 A014855 A298479

Adjacent sequences: A352086 A352087 A352088 * A352090 A352091 A352092

Keyword

nonn,base

Author

Amiram Eldar, Mar 04 2022

Status

approved