A328216 Weak-Fibonacci-Niven numbers: numbers divisible by the number of terms in at least one representation as a sum of distinct Fibonacci numbers.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 18, 21, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 55, 56, 57, 58, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 85, 89, 90, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 108

Offset

1,2

Comments

The Fibonacci numbers F(1) = F(2) = 1 can be used at most once in the representation.

Grundman proved that there are infinitely many runs of 6 consecutive weak-Fibonacci-Niven numbers by showing that if m = F(240k) + F(14) + F(9) for k >= 1, then m, m+1, ... m+5 are 6 consecutive weak-Fibonacci-Niven numbers.

Links

Table of n, a(n) for n=1..67.

Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Example

6 is in the sequence since it can be represented as the sum of 2 Fibonacci numbers, 1 + 5, and 2 is a divisor of 6.

Mathematica

m = 10; v = Array[Fibonacci, m, 2]; vm = v[[-1]]; seq = {}; Do[s = Subsets[v, 2^m, {k}]; If[(sum = Total @@ s) <= vm && Divisible[sum, Length @@ s], AppendTo[seq, sum]] , {k, 2, 2^m}]; Union @ seq

Crossrefs

Supersequence of A328208 and A328212.

Cf. A000045, A005349.

Sequence in context: A192188 A039268 A039162 * A071959 A176845 A196127

Adjacent sequences: A328213 A328214 A328215 * A328217 A328218 A328219

Keyword

nonn

Author

Amiram Eldar, Oct 07 2019

Status

approved