A053029 Numbers with 4 zeros in Fibonacci numbers mod m.

5, 10, 13, 17, 25, 26, 34, 37, 50, 53, 61, 65, 73, 74, 85, 89, 97, 106, 109, 113, 122, 125, 130, 137, 146, 149, 157, 169, 170, 173, 178, 185, 193, 194, 197, 218, 221, 226, 233, 250, 257, 265, 269, 274, 277, 289, 293, 298, 305, 313, 314, 317, 325, 337, 338, 346

Offset: 1

Author

Henry Bottomley, Feb 23 2000

Keywords

• nonn

Comments

Conjecture: m is on this list iff m is an odd number all of whose factors are on this list or m is twice such an odd number.

A001176(a(n)) = A128924(a(n),1) = 4. - Reinhard Zumkeller, Jan 17 2014

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.
• Brennan Benfield and Oliver Lippard, Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers, arXiv:2407.20048 [math.NT], 2024.
• M. Renault, Fibonacci sequence modulo m

Crossrefs

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.

| m=1 | m=2 | m=3

-----------------------------+----------+---------+---------

The sequence {x(n)} | A000045 | A000129 | A006190

The sequence {w(k)} | A001176 | A214027 | A322906

Primes p such that w(p) = 1 | A112860* | A309580 | A309586

Primes p such that w(p) = 2 | A053027 | A309581 | A309587

Primes p such that w(p) = 4 | A053028 | A261580 | A309588

Numbers k such that w(k) = 1 | A053031 | A309583 | A309591

Numbers k such that w(k) = 2 | A053030 | A309584 | A309592

Numbers k such that w(k) = 4 | this seq | A309585 | A309593

* and also A053032 U {2}

Programs

Haskell

a053029 n = a053029_list !! (n-1)
a053029_list = filter ((== 4) . a001176) [1..]
-- Reinhard Zumkeller, Jan 17 2014