A389401 Composite numbers that divide at least one term in every Fibonacci sequence.

4, 6, 9, 14, 27, 49, 81, 86, 98, 134, 206, 243, 254, 326, 343, 446, 529, 566, 686, 729, 734, 926, 974, 1046, 1094, 1214, 1286, 1454, 1574, 1646, 1766, 1814, 1849, 2126, 2187, 2246, 2401, 2606, 2654, 2846, 2894, 3086, 3134, 3254, 3326, 3446, 3494, 3566

Offset

1,1

Comments

Here a Fibonacci sequence is a sequence which begins with any two integers and continues using the rule s(n+2) = s(n+1) + s(n). These composite numbers divide at least one number in each such sequence. See similar comment in A000057.

For a number to be in this sequence, all its prime factors must be in A000057.

This sequence can apparently be generated from A000057 (the primes of A064414):

With the exception of 6, which is in the sequence but doesn't follow the following rules, for each term x taken from A000057:

- if 2*x == 14 or == 86 mod 120, all numbers of the form 2*x^k are part of the sequence, for k>0

- with the exceptions of 8 and its multiples, all x^k are part of the sequence for all k>1

Example: for x=2: 4 = x^2 is in the sequence, but because 4 is not congruent to 14 or 86 mod 120, 8 is not in the sequence.

Links

Daniel Mondot, Table of n, a(n) for n = 1..10000

Example

23 is prime as is part of A000057.
529 = 23^2 is composite and is part of this sequence.

Crossrefs

Subsequence of A064414.

Cf. A000057, A232656.

Sequence in context: A137371 A179463 A086697 * A372533 A374739 A287568

Adjacent sequences: A389398 A389399 A389400 * A389402 A389403 A389404

Keyword

nonn

Author

Daniel Mondot, Oct 02 2025

Status

approved