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

1,1

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

References

Benfield, Brennan, and Oliver Lippard. "Connecting zeros in Pisano periods to prime factors of K-Fibonacci numbers." The Fibonacci Quarterly 63.2 (2025): 240-258.

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

Prog

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

Crossrefs

For a list of sequences related to the numbers of zeros in a fundamental period of {x(n)}, where {x(n)} is a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n), see A053032.

Sequence in context: A288742 A113826 A376588 * A189822 A051205 A168114

Adjacent sequences: A053027 A053028 A053029 * A053031 A053032 A053033

Keyword

nonn

Author

Henry Bottomley, Feb 23 2000

Status

approved