A053030 Numbers with 2 zeros in Fibonacci numbers mod m.

3, 6, 7, 8, 9, 12, 14, 15, 16, 18, 20, 21, 23, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 60, 63, 64, 66, 67, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 83, 84, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 103, 104

Offset: 1

Author

Henry Bottomley, Feb 23 2000

Keywords

• nonn

Comments

m is on this list iff m does not have 1 or 4 zeros in the Fibonacci sequence modulo m.

A001176(a(n)) = A128924(a(n),1) = 2. - Reinhard Zumkeller, Jan 17 2014

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Brennan Benfield and Oliver Lippard, Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers, arXiv:2407.20048 [math.NT], 2024. See p. 2.
• Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.
• 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 | this seq | A309584 | A309592

Numbers k such that w(k) = 4 | A053029 | A309585 | A309593

* and also A053032 U {2}

Programs

Haskell

a053030 n = a053030_list !! (n-1)
a053030_list = filter ((== 2) . a001176) [1..]
-- Reinhard Zumkeller, Jan 17 2014