A113649 -id:A113649 - cp's OEIS frontend

A113650 Fibonacci(p-J(p,5)) mod p^2, where p is the n-th prime and J is the Jacobi symbol.

2, 3, 5, 21, 55, 39, 272, 57, 345, 754, 775, 481, 1599, 1677, 752, 1484, 590, 2928, 469, 3905, 4234, 3871, 1743, 445, 3589, 9797, 2266, 2568, 2834, 6780, 1651, 8384, 7946, 16263, 17880, 9060, 6908, 26080, 7348, 22490, 31146, 23711, 17954, 5983

Offset: 1

Eric W. Weisstein, Nov 03 2005

nonn

A value of 0 indicates a Wall-Sun-Sun prime. No such prime is currently known. - Felix Fröhlich, Jun 07 2014

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Wall-Sun-Sun Prime

Cf. A113649.

a[n_]:=  ( p=Prime[n];Mod[Fibonacci[p-JacobiSymbol[p, 5]], Power[p, 2]]); Table[a[n], {n,1,50}] (* Javier Rivera Romeu, Mar 03 2022 *)

a(n)=my(p=prime(n));lift(Mod([1,1;1,0]^(p-kronecker(p,5)),p^2)[1,2]) \\ Charles R Greathouse IV, Oct 31 2011

def a(n):
    p = Primes().unrank(n-1)
    return fibonacci(p-jacobi_symbol(p, 5))%pow(p, 2)
for n in range(1, 100): print(a(n), end=", ") # Javier Rivera Romeu, Mar 04 2022

A271782 Smallest n-Wall-Sun-Sun prime.

Original entry on oeis.org

13, 241, 2, 3, 191, 5, 2, 3, 2683

Offset: 2

Felix Fröhlich, Apr 18 2016

A prime p is a k-Wall-Sun-Sun prime iff p^2 divides F_k(pi_k(p)), where F_k(n) is the k-Fibonacci numbers, i.e., a Lucas sequence of first kind with (P,Q) = (k,-1), and pi_k(p) is the Pisano period of k-Fibonacci numbers modulo p (cf. A001175, A175181-A175185).
For prime p > 2 not dividing k^2 + 4, it is a k-Wall-Sun-Sun prime iff p^2 | F_k(p-((k^2+4)/p)), where ((k^2+4)/p) is the Kronecker symbol.
a(1) would be the smallest Wall-Sun-Sun prime whose existence is an open question.
a(12)..a(16) = 2, 3, 3, 29, 2. a(18)..a(33) = 3, 11, 2, 23, 3, 3, 2, 5, 79, 3, 2, 7, 23, 3, 2, 239. a(36)..a(38) = 2, 7, 17. a(40), a(41) = 2, 3. a(43)..a(46) = 5, 2, 3, 41. - R. J. Mathar, Apr 22 2016
a(17) = 1192625911, a(35) = 153794959, a(39) = 30132289567, a(47)..a(50) = 139703, 2, 3, 3. If they exist, a(11), a(34), a(42) are greater than 10^12. - Max Alekseyev, Apr 26 2016
Column k = 1 of table T(n, k) of k-th n-Wall-Sun-Sun prime (that table is not yet in the OEIS as a sequence). - Felix Fröhlich, Apr 25 2016
From Richard N. Smith, Jul 16 2019: (Start)
a(n) = 2 if and only if n is divisible by 4.
a(n) = 3 if and only if n == 5, 9, 13, 14, 18, 22, 23, 27, 31 (mod 36). (End)

Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41:1 (2009), 497-504.
Richard N. Smith, Table of n, a(n) for n = 1..1024 (or 0 if unknown, searched up to 2^22)
Richard N. Smith, List of odd n-Wall-Sun-Sun primes <= 2^20 for 1<=n<=1024
Wikipedia, Wall-Sun-Sun prime

Cf. A001177, A039951, A113649, A113650, A113651, A214028, A237517, A237835, A241014, A244801, A253247, A268478.

A271782(k) = forprime(p=2,10^8, if( (([0,1;1,k]*Mod(1,p^2))^(p-kronecker(k^2+4,p)))[1,2]==0, return(p);); ); \\ Max Alekseyev, Apr 22 2016, corrected by Richard N. Smith, Jul 16 2019 to include p=2 and p divides k^2+4

a(4n) = 2.

Edited by Max Alekseyev, Apr 25 2016

cp's OEIS Frontend

A113650 Fibonacci(p-J(p,5)) mod p^2, where p is the n-th prime and J is the Jacobi symbol.

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