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A053029
Numbers with 4 zeros in Fibonacci numbers mod m.
21
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
1,1
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
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.
PROG
(Haskell)
a053029 n = a053029_list !! (n-1)
a053029_list = filter ((== 4) . 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 | A053030 | A309584 | A309592
Numbers k such that w(k) = 4 | this seq | A309585 | A309593
* and also A053032 U {2}
Sequence in context: A313394 A313395 A134961 * A313396 A282740 A313397
KEYWORD
nonn
AUTHOR
Henry Bottomley, Feb 23 2000
STATUS
approved