A045468 -id:A045468 - cp's OEIS frontend

A069107 Composite numbers k that divide Fibonacci(k+1).

323, 377, 2834, 3827, 5777, 6479, 10877, 11663, 18407, 19043, 20999, 23407, 25877, 27323, 34943, 35207, 39203, 44099, 47519, 50183, 51983, 53663, 60377, 65471, 75077, 78089, 79547, 80189, 81719, 82983, 84279, 84419, 86063, 90287, 94667

Offset: 1

Benoit Cloitre, Apr 06 2002

Primes p congruent to +2 or -2 (mod 5) divide Fibonacci(p+1) (cf. A003631 and [Hardy and Wright]).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, Chap. X, p. 150.

Chai Wah Wu, Table of n, a(n) for n = 1..2000 (n = 1..250 from Reinhard Zumkeller, n = 251..1000 from Giovanni Resta)

Cf. A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

Cf. A000045, A010051, A023172, A069104.

a069107 n = a069107_list !! (n-1)
a069107_list = h 2 $ drop 3 a000045_list where
   h n (fib:fibs) = if fib `mod` n > 0 || a010051 n == 1
       then h (n+1) fibs else n : h (n+1) fibs
-- Reinhard Zumkeller, Oct 13 2011

Select[Range[2,100000],!PrimeQ[#]&&Divisible[Fibonacci[#+1],#]&] (* Harvey P. Dale, Sep 18 2011 *)

is(n)=((Mod([1,1;1,0],n))^(n+1))[1,2]==0 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Oct 07 2016

Fibonacci(2*a(n)) mod a(n) = a(n) - 1. - Gary Detlefs, May 26 2014

Corrected by Ralf Stephan, Oct 17 2002

A042993 Primes congruent to {0, 2, 3} mod 5.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 433, 443, 457

Offset: 1

N. J. A. Sloane

Also, primes p that are quadratic nonresidues modulo 5 (and from the quadratic reciprocity law, odd p such that 5 is a quadratic nonresidue modulo p). For primes p' that are quadratic residues modulo 5 (and such that 5 is a quadratic residue mod p') see A045468. - Lekraj Beedassy, Jul 13 2004

Primes p that divide Fibonacci(p+1). - Ron Knott, Jun 27 2014

			For prime 7, Fibonacci(8) = 21 = 3*7, for prime 13, Fibonacci(14) = 377 = 13*29.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, Theorem 180

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes dividing A001654.

Cf. A038872 for primes p which divide Fibonacci(p-1). - Ron Knott, Jun 27 2014

[p: p in PrimesUpTo(600) | p mod 5 in [0, 2, 3]]; // Vincenzo Librandi, Aug 09 2012

Select[Prime[Range[100]],MemberQ[{0,2,3},Mod[#,5]]&] (* Harvey P. Dale, Mar 03 2012 *)

A057538 Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.

Original entry on oeis.org

1, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 91, 101, 109, 119, 121, 131, 139, 149, 151, 161, 169, 179, 181, 191, 199, 209, 211, 221, 229, 239, 241, 251, 259, 269, 271, 281, 289, 299, 301, 311, 319, 329, 331, 341, 349, 359, 361, 371, 379, 389, 391, 401, 409

Offset: 1

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

Also numbers congruent to +-1 or +-11 modulo 30 and numbers k where (k^2 - 1)/120 is an integer; all but the first two prime legs of Pythagorean triangles which also have prime hypotenuses appear within in this sequence (A048161). - Henry Bottomley, Jan 31 2002
Numbers k such that k^2 == 1 (mod 30). - Gary Detlefs, Apr 16 2012
Subsequence of primes gives A045468. - Ray Chandler, Jul 29 2019

			229 is congruent to 1 (mod 2), 1 (mod 3), 1 (mod 4) and -1 (mod 5).
x+ 11*x^2 + 19*x^3 + 29*x^4 + 31*x^5 + 41*x^6 + 49*x^7 + 59*x^8 + 61*x^9 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
A. Feist, Maple source for birthday sets. [Broken link]
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A007310, A045468, A057539, A057540, A057541, A093722.

for n from 1 to 409 do if (n^2 mod 30 =1) then print(n) fi od; # Gary Detlefs, Apr 17 2012

a057538[n_] := Block[{f},
  f[x_] :=
   If[Mod[x, #] == 1 || Mod[x, #] == # - 1, True, False] & /@
    Range[2, 5];
Select[Range[n], DeleteDuplicates[f[#]] == {True} &]]; a057538[409] (* Michael De Vlieger, Dec 26 2014 *)

{a(n+1) = (n\4*3 + n%4)*10 + (-1)^(n\2)} /* Michael Somos, Oct 17 2006 */

A093722(n) = (a(n)^2 - 1)/120.
G.f.: x * (1 + 10*x + 8*x^2 + 10*x^3 + x^4) / ((1 - x) * (1 - x^4)). a(-1 - n) = -a(n). - Michael Somos, Jan 21 2012
4*a(n) = 30*(n+1) - 45 + 5*(-1)^n + 6*(-1)^floor((n+1)/2). - R. J. Mathar, Jul 30 2019

Corrected by Henry Bottomley, Jan 31 2002

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A069106 Composite numbers k such that k divides F(k-1) where F(j) are the Fibonacci numbers.

Original entry on oeis.org

442, 1891, 2737, 4181, 6601, 6721, 8149, 13201, 13981, 15251, 17119, 17711, 30889, 34561, 40501, 51841, 52701, 64079, 64681, 67861, 68101, 68251, 78409, 88601, 88831, 90061, 96049, 97921, 115231, 118441, 138601, 145351, 146611, 150121, 153781, 163081, 179697, 186961, 191351, 194833

Offset: 1

Benoit Cloitre, Apr 06 2002

Primes p congruent to 1 or 4 (mod 5) divide F(p-1) (cf. A045468 and [Hardy and Wright]).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, Chap. X, p. 150.

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

Subsequence of A123976.
Cf. A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).
Cf. A002808, A000045.

#include  #include  #define STARTN 10 #define N_OF_MILLER_RABIN_TESTS 5 int main() { mpz_t n, f1, f2; int flag=0; /* flag? 0: f1 contains current F[n-1] 1: f2 = F[n-1] */ mpz_set_ui (n, STARTN); mpz_init (f1); mpz_init (f2); mpz_fib2_ui (f1, f2, STARTN-1); for (;;) { if (mpz_probab_prime_p (n, N_OF_MILLER_RABIN_TESTS)) goto next_iter; if (mpz_divisible_p (!flag? f1:f2, n)) { mpz_out_str (stdout, 10, n); printf (" "); fflush (stdout); } next_iter: mpz_add_ui (n, n, 1); mpz_add (!flag? f2:f1, f1, f2); flag = !flag; } }

a069106 n = a069106_list !! (n-1)
a069106_list = [x | x <- a002808_list, a000045 (x-1) `mod` x == 0]
-- Reinhard Zumkeller, Jul 19 2013

A069106[nn_] := Select[Complement[Range[2,nn],Prime[Range[2,PrimePi[ nn]]]],Divisible[ Fibonacci[ #-1],#]&] (* Harvey P. Dale, Jul 05 2011 *)

fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
is(n)=!isprime(n) && !fibmod(n-1,n) && n>1 \\ Charles R Greathouse IV, Oct 06 2016

Corrected and extended (with C program) by Ralf Stephan, Oct 13 2002

a(35)-a(40) added by Reinhard Zumkeller, Jul 19 2013

A385192 Primes p such that multiplicative order of 5 modulo p is odd.

Original entry on oeis.org

2, 11, 19, 31, 59, 71, 79, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 239, 251, 269, 271, 311, 331, 359, 379, 389, 401, 409, 419, 431, 439, 461, 479, 491, 499, 541, 569, 571, 599, 619, 631, 659, 691, 719, 739, 751, 811, 829, 839, 859, 911, 919, 941, 971, 991

Offset: 1

Jianing Song, Jun 20 2025

The multiplicative order of 5 modulo a(n) is A385193(n).
Contained in primes congruent to 1 or 4 modulo 5 (primes p such that 5 is a quadratic residue modulo p, A045468), and contains primes congruent to 11 or 19 modulo 20 (A122869).
Conjecture: this sequence has density 1/3 among the primes.

			101 is a term since 5^25 == 1 (mod 101).

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A040105, which (without the terms 2 and 5) is itself a subsequence of A045468.
Contains A122869 as a proper subsequence.
Cf. A385193 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), this sequence (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[5, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385192(p) = isprime(p) && (p!=5) && znorder(Mod(5,p))%2

A054108 a(n) = (-1)^(n+1)Sum_{k=0..n+1}(-1)^kbinomial(2*k,k).

Original entry on oeis.org

1, 5, 15, 55, 197, 727, 2705, 10165, 38455, 146301, 559131, 2145025, 8255575, 31861025, 123256495, 477823895, 1855782325, 7219352975, 28125910825, 109720617995, 428537256445, 1675561707275, 6557869020325, 25689734662775

Offset: 0

Clark Kimberling

nonn

p divides a((p-3)/2) for p in A045468 (primes congruent to {1, 4} mod 5). - Alexander Adamchuk, Jul 05 2006
The sequence 1,1,5,15,55,... has general term sum{k=0..n, (-1)^(n-k)*C(2k,k)}. Its Hankel transform is A082761. - Paul Barry, Apr 10 2007
From Paul Barry, Mar 29 2010: (Start)
The sequence 1,1,5,15,... has g.f. 1/((1+x)*sqrt(1-4x)).
The doubled sequence 1,1,1,1,5,5,... has e.g.f. dif(int((sin(x-t)+cos(x-t))*Bessel_I(0,2t),t,0,x),x). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

T(2n, n), array T as in A054106.

Cf. A066796, A000984, A054109, A006134, A045468.

Table[Sum[(-1)^(k+n)*((2k)!/(k!)^2),{k,0,n}], {n,1,50}] (* Alexander Adamchuk, Jul 05 2006 *)
CoefficientList[Series[(1/Sqrt[1-4*x]/(1+x)-1)/x, {x, 0, 20}], x]
(* or *)
Table[(-1)^(n+1)*Sum[(-1)^k*Binomial[2*k, k], {k, 0, n+1}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2012 *)
Round@Table[Binomial[2 (n + 2), n + 2] Hypergeometric2F1[1, n + 5/2, n + 3, -4] - (-1)^n/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

a(n)=(-1)^(n+1)*sum(k=0,n+1,(-1)^k*binomial(2*k,k))

from math import comb
def A054108(n): return (1 if n % 2 else -1)*sum((-1 if k % 2 else 1)*comb(2*k,k) for k in range(n+2)) # Chai Wah Wu, Jan 19 2022

a(n) = C(2n, n) - a(n-1) with a(0)=1. - Labos Elemer, Apr 26 2003
C(2n,n) - C(2n-2,n-1) + ... +(-1)^(k+n)*C(2k,k)+ ... + (-1)^(1+n)*C(2,1) + (-1)^n*C(0,0), where C(2k,k)=(2k)!/(k!)^2 - central binomial coefficients A000984[k]. - Alexander Adamchuk, Jul 05 2006
a(n) = Sum_{k=0..n} (-1)^(k+n)*((2k)!/(k!)^2). - Alexander Adamchuk, Jul 05 2006
G.f.: (1/sqrt(1-4*x)/(1+x)-1)/x = (-1 + 2/(U(0)-2*x))/(1+x) where U(k)= 2*(2*k+1)*x + (k+1) - 2*(k+1)*(2*k+3)*x/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jun 27 2012
a(n) ~ 2^(2*n+4)/(5*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 06 2012
Recurrence: (n+1)*a(n) = (3*n+1)*a(n-1) + 2*(2*n+1)*a(n-2). - Vaclav Kotesovec, Nov 06 2012

Formula from Benoit Cloitre, Sep 29 2002

Definition corrected by Vaclav Kotesovec, Nov 06 2012

A123976 Numbers k such that Fibonacci(k-1) is divisible by k.

Original entry on oeis.org

1, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 442, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601

Offset: 1

Tanya Khovanova, Oct 30 2006

nonn

a(n) is a union of {1}, A069106(n) and A045468(n). Composite a(n) are listed in A069106(n) = {442, 1891, 2737, 4181, 6601, 6721, 8149, ...}. Prime a(n) are listed in A045468(n) = {11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, ...} Primes congruent to {1, 4} mod 5. - Alexander Adamchuk, Nov 02 2006

			Fibonacci(10) = 55, is divisible by 11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A069106, A045468, A069104, A069107, A003631, A000045, A023172, A159051.

import Data.List (elemIndices)
a123976 n = a123976_list !! (n-1)
a123976_list = map (+ 1) $ elemIndices 0 $ zipWith mod a000045_list [1..]
-- Reinhard Zumkeller, Oct 13 2011

Select[Range[1000], IntegerQ[Fibonacci[ # - 1]/# ] &]

is(n)=((Mod([1,1;1,0],n))^n)[2,2]==0 \\ Charles R Greathouse IV, Feb 03 2014

A341783 Absolute values of norms of prime elements in Z[(1+sqrt(5))/2], the ring of integers of Q(sqrt(5)).

Original entry on oeis.org

4, 5, 9, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 169, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 289, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 529

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(5))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 0, 1, 4 modulo 5 and the squares of primes congruent to 2, 3 modulo 5.
For primes p == 1, 4 (mod 5), there are two distinct ideals with norm p in Z[(1+sqrt(5))/2], namely (x + y*(1+sqrt(5))/2) and (x + y*(1-sqrt(5))/2), where (x,y) is a solution to x^2 + x*y - y^2 = p; for p = 5, (sqrt(5)) is the unique ideal with norm p; for p == 2, 3 (mod 5), (p) is the only ideal with norm p^2.

			norm((7 + sqrt(5))/2) = norm((7 - sqrt(5))/2) = 11;
norm((9 + sqrt(5))/2) = norm((9 - sqrt(5))/2) = 19;
norm((11 + sqrt(5))/2) = norm((11 - sqrt(5))/2) = 29;
norm(6 + sqrt(5)) = norm(6 - sqrt(5)) = 31.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A080891, A038872, A045468, A003631.
The number of nonassociative elements with absolute value of norm n (also the number of distinct ideals with norm n) is given by A035187.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), this sequence (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341783(n) = my(disc=5); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

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

A119996 Numerator of Sum_{k=1..n} 1/(Fibonacci(k)*Fibonacci(k+2)).

Original entry on oeis.org

1, 5, 14, 39, 103, 272, 713, 1869, 4894, 12815, 33551, 87840, 229969, 602069, 1576238, 4126647, 10803703, 28284464, 74049689, 193864605, 507544126, 1328767775, 3478759199, 9107509824, 23843770273, 62423800997, 163427632718

Offset: 1

Alexander Adamchuk, Aug 03 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A000045, A059248, A064831, A001654, A045468, A064739, A091729.

F:=Fibonacci;; List([1..30], n-> F(n+1)*F(n+2)-1); # G. C. Greubel, Jul 23 2019

[Fibonacci(n+1)* Fibonacci(n+2)-1: n in [1..30]]; // Vincenzo Librandi, Aug 14 2012

with(combinat): seq(fibonacci(n+1)*fibonacci(n+2)-1, n=1..30); # Zerinvary Lajos, Jan 31 2008

Numerator[Table[Sum[1/(Fibonacci[k]*Fibonacci[k+2]),{k,n}],{n,30}]]
LinearRecurrence[{3,0,-3,1},{1,5,14,39},30] (* Harvey P. Dale, Aug 22 2011 *)

vector(30, n, f=fibonacci; f(n+1)*f(n+2)-1) \\ G. C. Greubel, Jul 23 2019

f=fibonacci; [f(n+1)*f(n+2)-1 for n in (1..30)] # G. C. Greubel, Jul 23 2019

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4); a(0)=1, a(1)=5, a(2)=14, a(3)=39. - Harvey P. Dale, Aug 22 2011
G.f.: ((x-2)*x-1)/(x^4 - 3*x^3 + 3*x - 1). - Harvey P. Dale, Aug 22 2011
a(n) = Fibonacci(n+1)*Fibonacci(n+2) - 1. - Gary Detlefs, Mar 31 2012
a(n) = Sum_{k=1..n} Fibonacci(k+1)^2. Can be proved by induction from Gary Detlefs formula. - Joel Courtheyn, Mar 15 2021

cp's OEIS Frontend

A069107 Composite numbers k that divide Fibonacci(k+1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A042993 Primes congruent to {0, 2, 3} mod 5.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

A057538 Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A069106 Composite numbers k such that k divides F(k-1) where F(j) are the Fibonacci numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

C

Haskell

Mathematica

PARI

Extensions

A385192 Primes p such that multiplicative order of 5 modulo p is odd.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A054108 a(n) = (-1)^(n+1)*Sum_{k=0..n+1}(-1)^k*binomial(2*k,k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A123976 Numbers k such that Fibonacci(k-1) is divisible by k.

A054108 a(n) = (-1)^(n+1)Sum_{k=0..n+1}(-1)^kbinomial(2*k,k).