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A053030
Numbers with 2 zeros in Fibonacci numbers mod m.
21
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
1,1
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
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.
PROG
(Haskell)
a053030 n = a053030_list !! (n-1)
a053030_list = filter ((== 2) . a001176) [1..]
-- Reinhard Zumkeller, Jan 17 2014
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}
Sequence in context: A288742 A113826 A376588 * A189822 A051205 A168114
KEYWORD
nonn
AUTHOR
Henry Bottomley, Feb 23 2000
STATUS
approved