A094400 -id:A094400 - cp's OEIS frontend

A094394 Odd composites m that divide Fibonacci(m)-1.

323, 2737, 4181, 6479, 6721, 7743, 11663, 13201, 15251, 18407, 19043, 23407, 27071, 34561, 34943, 35207, 39203, 44099, 47519, 51841, 51983, 53663, 54839, 64079, 64681, 65471, 67861, 68251, 72831, 78089, 79547, 82983, 86063, 90061, 94667

Offset: 1

Eric Rowland, May 01 2004

nonn

No terms satisfy the Fermat criterion 2^(a(n)-1) mod a(n) = 1. - Gary Detlefs, May 25 2014

For each prime p, Fibonacci(p) = 5^((p-1)/2) mod p, so p divides Fibonacci(p) - 1 for each prime p=10k+-1. Hence it is interesting to seek also nonprimes with the same property, a motivation for this sequence. - Robert FERREOL, Jul 14 2015

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A094395, A094400.

with(combinat):test:=n->(fibonacci(n)-1) mod n= 0:
select(test and not isprime ,[seq(2*k+1,k=1..10000)]); # Robert FERREOL, Jul 14 2015

Select[Range[2, 50000], OddQ[#] && ! PrimeQ[#] && Mod[Fibonacci[#] - 1, #] == 0 &]

main(m)=forcomposite(n=1,m,if(((n%2==1)&&(fibonacci(n)-1)%n==0),print1(n,", "))); \\ Anders Hellström, Aug 12 2015

Offset corrected by Giovanni Resta, Jul 20 2013

A094401 Composite n such that n divides both Fibonacci(n-1) and Fibonacci(n) - 1.

Original entry on oeis.org

2737, 4181, 6721, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 90061, 96049, 97921, 118441, 146611, 163081, 179697, 186961, 194833, 197209, 219781, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 327313, 330929

Offset: 1

Eric Rowland, May 01 2004

nonn

Composite n such that Q^(n-1) = I (mod n), where Q is the Fibonacci matrix {{1,1},{1,0}} and I is the identity matrix. The identity is also true for the primes congruent to 1 or 4 (mod 5), which is sequence A045468. The period of Q^k (mod n) is the same as the period of the Fibonacci numbers F(k) (mod n), A001175. Hence the terms in this sequence are the composite n such that A001175(n) divides n-1. [T. D. Noe, Jan 09 2009]

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)
Eric Weisstein, MathWorld: Fibonacci Q-Matrix.

Cf. A005845, A069106, A094394, A094400.

Select[Range[2, 50000], ! PrimeQ[ # ] && Mod[Fibonacci[ # - 1], # ] == 0 && Mod[Lucas[ # ] - 1, # ] == 0 &]

More terms from Ryan Propper, Sep 24 2005

A094402 Numbers k that divide both Fibonacci(k+1) and Lucas(k) + 1.

Original entry on oeis.org

1, 2, 323, 6479, 11663, 18407, 19043, 23407, 34943, 35207, 39203, 44099, 47519, 51983, 53663, 65471, 78089, 79547, 82983, 86063, 94667, 104663, 109871, 121103, 139359, 142883, 157079, 168299, 195227, 196559, 200147, 233519

Offset: 1

Eric Rowland, May 01 2004

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A069107, A094398, A094400.

Select[Range[10^5],Divisible[Fibonacci[#+1],#]&&Divisible[LucasL[#]+1,#]&] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

is(n)=(Mod([0,1;1,1],n)^(n+1)*[0;1])[1,1]==0 && (Mod([0,1;1,1],n)^n*[2;1])[1,1]==-1 \\ Charles R Greathouse IV, Nov 04 2016

More terms from Emeric Deutsch, Apr 13 2005

A095401 number of steps required to reach 1 if the following modified juggler map is iterated: a[n]=(1-Mod[n, 2])Floor[n^(3/4)]+Mod[n, 2]Floor[n^(4/3)]; original exponents {1/2, 3/2} are replaced with {3/4, 4/3}.

Original entry on oeis.org

0, 1, 3, 2, 4, 4, 8, 3, 5, 5, 7, 5, 7, 9, 11, 4, 8, 4, 6, 6, 12, 6, 18, 6, 14, 8, 12, 6, 14, 6, 18, 8, 18, 10, 12, 10, 16, 12, 14, 12, 20, 5, 7, 9, 11, 9, 13, 5, 9, 5, 9, 7, 13, 7, 11, 7, 11, 13, 19, 13, 15, 7, 9, 7, 17, 19, 21, 19, 23, 7, 11, 7, 13, 15, 17, 15, 19, 9, 11, 9, 11, 13, 15, 13, 19

Offset: 1

Labos Elemer, Jun 18 2004

nonn

			n=101: the trajectory is {101, 470, 100, 31, 97, 445, 3397, 51065, 1894513, 234421146, 1894512, 51064, 3396, 444, 96, 30, 12, 6, 3, 4, 2, 1}, number of required steps is a[101]=22-1=21.

Cf. A007320, A094683, A094716, A094396-A094400.

e[x_]:=e[x]=(1-Mod[x, 2])*Floor[N[x^(3/4), 50]] +Mod[x, 2]*Floor[N[x^(4/3), 50]];e[1]=1; fe[x_]:=Delete[FixedPointList[e, x], -1]; Table[ -1+Length[fe[w]], {w, 1, 150}]

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A094394 Odd composites m that divide Fibonacci(m)-1.

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A094401 Composite n such that n divides both Fibonacci(n-1) and Fibonacci(n) - 1.

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A094402 Numbers k that divide both Fibonacci(k+1) and Lucas(k) + 1.

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A095401 number of steps required to reach 1 if the following modified juggler map is iterated: a[n]=(1-Mod[n, 2])*Floor[n^(3/4)]+Mod[n, 2]*Floor[n^(4/3)]; original exponents {1/2, 3/2} are replaced with {3/4, 4/3}.

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A095401 number of steps required to reach 1 if the following modified juggler map is iterated: a[n]=(1-Mod[n, 2])Floor[n^(3/4)]+Mod[n, 2]Floor[n^(4/3)]; original exponents {1/2, 3/2} are replaced with {3/4, 4/3}.