A003631 -id:A003631 - cp's OEIS frontend

A094809 Numbers of the form Fibonacci(p+1)/p, where p are primes >= 7 that end in 3 or 7 (i.e., p = A003631(n) for n > 2).

3, 29, 152, 2016, 1056437, 16311831, 102287808, 1627690024, 1085424779823, 17876295136009, 1933742696582736, 1394864457161925217, 23571778916504612451, 155523623837575361328, 2642576343530770503704, 1982116737665744001184443, 225989772612884036918821112, 2983063546028389514347409745197

Offset: 1

Lekraj Beedassy, Jun 11 2004

nonn

Cf. A003631, A094808, A096028.

With[{prs=Select[Prime[Range[4,50]],MemberQ[{3,7},Mod[#,10]]&]},Table[Fibonacci[p+1]/p,{p,prs}]] (* Harvey P. Dale, Sep 26 2024 *)

forprime(p=7,10^3,if((p+3)%5>1,next);print1(fibonacci(p+1)/p,",")) \\ Max Alekseyev, Jan 12 2007

a(n) = A096028(n+1). - Jinyuan Wang, Feb 24 2020

More terms from Max Alekseyev, Jan 12 2007

More terms from Jinyuan Wang, Feb 24 2020

A096028 Numbers of the form (Fibonacci(p+1))/p, where p are primes ending in 3 or 7 (i.e., A003631).

Original entry on oeis.org

1, 3, 29, 152, 2016, 1056437, 16311831, 102287808, 1627690024, 1085424779823, 17876295136009, 1933742696582736, 1394864457161925217, 23571778916504612451, 155523623837575361328, 2642576343530770503704, 1982116737665744001184443, 225989772612884036918821112, 2983063546028389514347409745197

Offset: 1

Lekraj Beedassy and Robert G. Wilson v, Jun 15 2004

nonn

Essentially the same as A094809. - R. J. Mathar, Sep 04 2008

Cf. A094808, A094809.

s = Select[ Prime[ Range[35]], Mod[ #, 10] == 3 || Mod[ #, 10] == 7 &]; Fibonacci[s + 1]/s

forprime(p=3, 10^3, if((p+3)%5>1, next); print1(fibonacci(p+1)/p, ", ")); \\ Jinyuan Wang, Feb 24 2020

More terms from Jinyuan Wang, Feb 24 2020

A019546 Primes whose digits are primes; primes having only {2, 3, 5, 7} as digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557

Offset: 1

R. Muller

Intersection of A046034 and A000040; A055642(a(n)) = A193238(a(n)). - Reinhard Zumkeller, Jul 19 2011
Ribenboim mentioned in 2000 the following number as largest known term: 72323252323272325252 * (10^3120 - 1) / (10^20 - 1) + 1. It has 3120 digits, and was discovered by Harvey Dubner in 1992. Larger terms are 22557252272*R(15600)/R(10) and 2255737522*R(15600) where R(n) denotes the n-th repunit (see A002275): Both have 15600 digits and were found in 2002, also by Dubner (see Weisstein link). David Broadhurst reports in 2003 an even longer number with 82000 digits: (10^40950+1) * (10^20055+1) * (10^10374 + 1) * (10^4955 + 1) * (10^2507 + 1) * (10^1261 + 1) * (3*R(1898) + 555531001*10^940 - R(958)) + 1, see link. - Reinhard Zumkeller, Jan 13 2012
The smallest and largest primes that use exactly once the four prime decimal digits are respectively a(27)= 2357 and a(54) = 7523. - Bernard Schott, Apr 27 2023

Paulo Ribenboim, Prime Number Records (Chap 3), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 76.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
József Bölcsföldi and György Birkás, Golden ratio prime numbers, International Journal of Engineering Science Invention (2018) Vol. 6 Issue 12, 82-85.
David Broadhurst: primeform, 82000-digit prime with all digits prime [Broken link]
David Broadhurst, 82000-digit prime with all digits prime, digest of 2 messages in primeform Yahoo group, Oct 20 - Oct 25, 2003.
Chris K. Caldwell, The Prime Glossary: Prime-digit prime
Chris K. Caldwell and G. L. Honaker, Jr., 2357, Prime Curios!
Chris K. Caldwell and G. L. Honaker, Jr., 7523, Prime Curios!
H. Ibstedt, A Few Smarandache Integer Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, pp. 171-183.
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.
Eric Weisstein's MathWorld Headline News, Two Gigantic Primes with Prime Digits Found
Eric Weisstein's World of Mathematics, Smarandache Sequences
Index to entries for primes with digits in a given set

Cf. A045336, A003631, A034844, A179336, A109066, A215927.
Cf. A020463 (subsequence).
A093162, A093164, A093165, A093168, A093169, A093672, A093674, A093675, A093938 and A093941 are subsequences. - XU Pingya, Apr 20 2017

a019546 n = a019546_list !! (n-1)
a019546_list = filter (all (`elem` "2357") . show )
                      ([2,3,5] ++ (drop 2 a003631_list))
-- Or, much more efficient:
a019546_list = filter ((== 1) . a010051) $
                      [2,3,5,7] ++ h ["3","7"] where
   h xs = (map read xs') ++ h xs' where
     xs' = concat $ map (f xs) "2357"
     f xs d = map (d :) xs
-- Reinhard Zumkeller, Jul 19 2011

[p: p in PrimesUpTo(5600) | Set(Intseq(p)) subset [2,3,5,7]]; // Bruno Berselli, Jan 13 2012

Select[Prime[Range[700]], Complement[IntegerDigits[#], {2, 3, 5, 7}] == {} &] (* Alonso del Arte, Aug 27 2012 *)
Select[Prime[Range[700]], AllTrue[IntegerDigits[#], PrimeQ] &] (* Ivan N. Ianakiev, Jun 23 2018 *)
Select[Flatten[Table[FromDigits/@Tuples[{2,3,5,7},n],{n,4}]],PrimeQ] (* Harvey P. Dale, Apr 05 2025 *)

is_A019546(n)=isprime(n) & !setminus(Set(Vec(Str(n))),Vec("2357")) \\ M. F. Hasler, Jan 13 2012

print1(2); for(d=1,4, forstep(i=1,4^d-1,[1,1,2], p=sum(j=0,d-1,10^j*[2,3,5,7][(i>>(2*j))%4+1]); if(isprime(p), print1(", "p)))) \\ Charles R Greathouse IV, Apr 29 2015

from itertools import product
from sympy import isprime
A019546_list = [2,3,5,7]+[p for p in (int(''.join(d)+e) for l in range(1,5) for d in product('2357',repeat=l) for e in '37') if isprime(p)] # Chai Wah Wu, Jun 04 2021

More terms from Cino Hilliard, Aug 06 2006

Thanks to Charles R Greathouse IV and T. D. Noe for massive editing support.

A038872 Primes congruent to {0, 1, 4} mod 5.

Original entry on oeis.org

5, 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, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601, 619

Offset: 1

N. J. A. Sloane

Also odd primes p such that 5 is a square mod p: (5/p) = +1 for p > 5.
Primes of the form x^2 + x*y - y^2 (as well as of the form x^2 + 3*x*y + y^2), both with discriminant = 5 and class number = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1. [This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales, Jun 06 2008. R. J. Mathar proved on Jul 22 2008 that this coincides with the present sequence.]
Also primes of the form 5x^2 - y^2 (cf. A031363). - N. J. A. Sloane, May 30 2014
Also primes of the form x^2 + 4*x*y - y^2. Every binary quadratic primitive form of discriminant 20 or 5 has proper solutions for positive integers N given in A089270, including the present primes. Proof from computing the corresponding representative parallel primitive forms, which leads to x^2 - 5 == 0 (mod N) or x^2 + x - 1 == 0 (mod N) which have solutions precisely for these positive N values, including these primes. - Wolfdieter Lang, Jun 19 2019
For a Pythagorean triple a, b, c, these primes (and 2) are the possible prime factors of 2a + b, |2a - b|, 2b + a, and 2b - a. - J. Lowell, Nov 05 2011
The prime factors of A028387(n^2+3n+1). - Richard R. Forberg, Dec 12 2014
Except for p = 5, these are primes p that divide Fibonacci(p-1). - Dmitry Kamenetsky, Jul 27 2015
Apart from the first term, these are rational primes that decompose in the field Q[sqrt(5)]. For example, 11 = ((7 + sqrt(5))/2)*((7 - sqrt(5))/2), 19 = ((9 + sqrt(5))/2)*((9 - sqrt(5))/2). - Jianing Song, Nov 23 2018
The possible prime factors of x^2 - x - 1. - Charles R Greathouse IV, Mar 18 2022

Z. I. Borevich and I. R. Shafarevich, Number Theory.

T. D. Noe, Table of n, a(n) for n = 1..1000
C. Banderier, Calcul de (5/p)
Henri Darmon, Andrew Wiles's Marvelous Proof, Notices of the AMS (2017), Volume 64, Number 3 pp. 209-216. See p. 211.
Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A045468, A089270.
Cf. A038872 (d=5); A038873 (d=8); A068228, A141123 (d=12); A038883 (d=13). A038889 (d=17); A141111, A141112 (d=65).
Cf. A003631 (complement with respect to A000040).

Filtered(Concatenation([5],Flat(List([1..140],k->[5*k-1,5*k+1]))),IsPrime); # Muniru A Asiru, Nov 24 2018

[ p: p in PrimesUpTo(700) | p mod 5 in {0,1,4}]; // Vincenzo Librandi, Aug 21 2012

select(isprime, [5, seq(op([5*k-1,5*k+1]),k=1..1000)]); # Robert Israel, Dec 22 2014

Join[{5}, Select[Prime[Range[4, 100]], Mod[#, 5] == 1 || Mod[#, 5] == 4 &]] (* Alonso del Arte, Nov 27 2011 *)

forprime(p=2,1e3,if(kronecker(5,p)>=0,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A045468(n-1) for n > 1. - Robert Israel, Dec 22 2014

a(n) ~ 2n*log(n). - Charles R Greathouse IV, Nov 29 2016

Corrected and extended by Peter K. Pearson, May 29 2005

Edited by N. J. A. Sloane, Jul 28 2008 at the suggestion of R. J. Mathar

A045468 Primes congruent to {1, 4} mod 5.

Original entry on oeis.org

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, 449, 461, 479, 491

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(5)). - N. J. A. Sloane, Dec 26 2017
These are also primes p that divide Fibonacci(p-1). - Jud McCranie
Primes ending in 1 or 9. - Lekraj Beedassy, Oct 27 2003
Also primes p such that p divides 5^(p-1)/2 - 4^(p-1)/2. - Cino Hilliard, Sep 06 2004
Primes p such that the polynomial x^2-x-1 mod p has 2 distinct zeros. - T. D. Noe, May 02 2005
Same as A038872, apart from the term 5. - R. J. Mathar, Oct 18 2008
Appears to be the primes p such that p^6 mod 210 = 1. - Gary Detlefs, Dec 29 2011
Primes in A047209, also in A090771. - Reinhard Zumkeller, Jan 07 2012
Primes p such that p does not divide Sum_{i=1..p} Fibonacci(i)^2. The sum is A001654(p). - Arkadiusz Wesolowski, Jul 23 2012
Primes congruent to {1, 9} mod 10. Legendre symbol (5, a(n)) = +1. For prime 5 this symbol (5, 5) is set to 0, and (5, prime) = -1 for prime == {3, 7} (mod 10), given in A003631. - Wolfdieter Lang, Mar 05 2021

Hardy and Wright, An Introduction to the Theory of Numbers, Chap. X, p. 150, Oxford University Press, Fifth edition.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, Chocolate Numbers, arXiv:1509.06093 [math.CO], 2015.
Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, Chocolate Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.1.7.
Index to sequences related to decomposition of primes in quadratic fields

Cf. A030430, A030433, A064739, A038872, A010051, A003631.

Subsequence of A123976.

a045468 n = a045468_list !! (n-1)
a045468_list = [x | x <- a047209_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[ p: p in PrimesUpTo(1000) | p mod 5 in {1,4} ]; // Vincenzo Librandi, Aug 13 2012

for n from 1 to 500 do if(isprime(n)) and (n^6 mod 210=1) then print(n) fi od;  # Gary Detlefs, Dec 29 2011

lst={};Do[p=Prime[n];If[Mod[p,5]==1||Mod[p,5]==4,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Select[Prime[Range[200]],MemberQ[{1,4},Mod[#,5]]&] (* Vincenzo Librandi, Aug 13 2012 *)

list(lim)=select(n->n%5==1||n%5==4,primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

A047221 Numbers that are congruent to {2, 3} mod 5.

Original entry on oeis.org

2, 3, 7, 8, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 47, 48, 52, 53, 57, 58, 62, 63, 67, 68, 72, 73, 77, 78, 82, 83, 87, 88, 92, 93, 97, 98, 102, 103, 107, 108, 112, 113, 117, 118, 122, 123, 127, 128, 132, 133, 137, 138, 142, 143, 147, 148, 152, 153

Offset: 1

N. J. A. Sloane

Theorem: if 5^((n-1)/2) = -1 (mod n) then n == 2 or 3 (mod 5) (see Crandall and Pomerance).
Start with 2. The next number, 3, cannot be written as the sum of two of the previous terms. So 3 is in. 4=2+2, 5=2+3, 6=3+3, so these are not in. But you cannot obtain 7, so the next term is 7. And so on. - Fabian Rothelius, Mar 13 2001
Also numbers k such that k^2 == -1 (mod 5). - Vincenzo Librandi, Aug 05 2010
For any (t,s) < n, a(t)*a(s) != a(n) and a(t) - a(s) != a(n). - Anders Hellström, Jul 01 2015
These numbers appear in the product of a Rogers-Ramanujan identity. See A003106 also for references. - Wolfdieter Lang, Oct 29 2016

Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 3.24, p. 154.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A118015 (floor(n^2/5)).
Cf. A003631 (primes), A094214.
Partitions into: A003106, A219607.

a047221 n = 5 * ((n - 1) `div` 2) + 3 - n `mod` 2
a047221_list = 2 : 3 : map (+ 5) a047221_list
-- Reinhard Zumkeller, Nov 27 2012

[ n : n in [1..165] | n mod 5 eq 2 or n mod 5 eq 3 ];

{2,3}+#&/@(5 Range[0,30])//Flatten (* Harvey P. Dale, Jan 22 2023 *)

Vec(x*(2+x+2*x^2)/((1+x)*(1-x)^2) + O(x^80)) \\ Michel Marcus, Jun 30 2015

a(n) = 5*(n-1) - a(n-1) (with a(1)=2). - Vincenzo Librandi, Aug 05 2010
a(n) = (10*n - 3*(-1)^n - 5)/4.
G.f.: x*(2+x+2*x^2)/((1+x)*(1-x)^2).
a(n)^2 = 5*A118015(a(n)) + 4.
a(n) = 5 * (floor(n-1)/2) + 3 - n mod 2. - Reinhard Zumkeller, Nov 27 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/5. - Amiram Eldar, Dec 07 2021
E.g.f.: 2 + ((5*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 23 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1.
Product_{n>=1} (1 + (-1)^n/a(n)) = 1/phi (A094214). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Apr 08 2002

Closed formula, g.f. and link added by Bruno Berselli, Nov 28 2010

A064739 Primes p such that Fibonacci(p)-1 is divisible by p.

Original entry on oeis.org

2, 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, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601, 619

Offset: 1

Shane Findley and N. J. A. Sloane, Oct 17 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

{2} union A045468. Complement is A003631 minus {2}.

lst={};Do[p=Prime[n];If[Mod[(Fibonacci[p]-1),p]==0,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 03 2009 *)
Select[Prime[Range[150]],Divisible[Fibonacci[#]-1,#]&] (* Harvey P. Dale, Sep 24 2022 *)

forprime(p=2,700, if((fibonacci(p)-1)%p==0,print1(p,", ")))

{ n=0; for (m=1, 10^9, p=prime(m); if ((fibonacci(p) - 1)%p==0, write("b064739.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 24 2009

Presumably this consists of 2 together with the primes congruent to +-1 mod 5.

More terms from Klaus Brockhaus, Oct 18 2001

A023173 Numbers k such that Fibonacci(k) == 1 (mod k).

Original entry on oeis.org

1, 2, 11, 19, 22, 29, 31, 38, 41, 58, 59, 61, 62, 71, 79, 82, 89, 101, 109, 118, 122, 131, 139, 142, 149, 151, 158, 178, 179, 181, 191, 199, 202, 211, 218, 229, 239, 241, 251, 262, 269, 271, 278, 281, 298, 302, 311, 323, 331, 349, 358, 359, 362, 379, 382, 389, 398, 401

Offset: 1

David W. Wilson

Sequence contains 1, A064739(k) for all k>=0, 2*A064739(k) for k>1. - Benoit Cloitre, Apr 06 2002

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000045, A045468, A003631, A064739.

lst={};Do[If[Mod[Fibonacci[n],n]==1,AppendTo[lst,n]],{n,1,7!}];Take[Union[lst],5! ] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2009 *)
Select[Range[500],Mod[Fibonacci[#]-1,#]==0&] (* Harvey P. Dale, Sep 22 2021 *)

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

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A094809 Numbers of the form Fibonacci(p+1)/p, where p are primes >= 7 that end in 3 or 7 (i.e., p = A003631(n) for n > 2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A096028 Numbers of the form (Fibonacci(p+1))/p, where p are primes ending in 3 or 7 (i.e., A003631).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A019546 Primes whose digits are primes; primes having only {2, 3, 5, 7} as digits.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

Python

Extensions

A038872 Primes congruent to {0, 1, 4} mod 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

Extensions

A045468 Primes congruent to {1, 4} mod 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

A047221 Numbers that are congruent to {2, 3} mod 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI