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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).

%I A352089 #16 Sep 07 2024 03:21:41
%S A352089 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,
%T A352089 60,68,69,72,75,76,80,81,82,84,87,88,90,94,96,100,108,115,116,120,126,
%U A352089 128,129,132,135,136,138,140,149,150,156,162,168,174,176,177,180
%N 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).
%C A352089 Numbers k such that A278043(k) | k.
%C A352089 The positive tribonacci numbers (A000073) are all terms.
%C A352089 If k = A000073(A042964(m)) is an odd tribonacci number, then k+1 is a term.
%C A352089 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
%D A352089 Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.
%H A352089 Amiram Eldar, <a href="/A352089/b352089.txt">Table of n, a(n) for n = 1..10000</a>
%H A352089 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 A352089 6 is a term since its minimal tribonacci representation, A278038(6) = 110, has A278043(6) = 2 1's and 6 is divisible by 2.
%t A352089 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]
%Y A352089 Cf. A000073, A042964, A278038, A278043.
%Y A352089 Subsequences: A352090, A352091, A352092.
%Y A352089 Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719.
%K A352089 nonn,base
%O A352089 1,2
%A A352089 _Amiram Eldar_, Mar 04 2022