A241505 -id:A241505 - cp's OEIS frontend

A113649 Fibonacci(n-J(n,5)) mod n^2, where J is the Jacobi symbol.

0, 2, 3, 2, 5, 5, 21, 34, 21, 55, 55, 89, 39, 37, 160, 98, 272, 293, 57, 365, 150, 101, 345, 433, 25, 665, 696, 709, 754, 440, 775, 994, 883, 1090, 765, 1241, 481, 230, 1511, 1355, 1599, 257, 1677, 805, 20, 1382, 752, 289, 2275, 1525, 1414, 821, 1484

Offset: 1

Eric W. Weisstein, Nov 03 2005

nonn

a(n) == 0 for n > 1 iff either n is a Wall-Sun-Sun prime (when n is prime) or a 'Wall-Sun-Sun pseudoprime' (when n is composite). The numbers meeting the second criterion are those composites where the congruence in A241505 is satisfied modulo n^2. No members are known from either of those two sets of numbers. - Felix Fröhlich, May 15 2015

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

Cf. A113650.

[Fibonacci(n-(KroneckerSymbol(n,5))) mod n^2: n in [1..70]]; // Vincenzo Librandi, May 16 2015

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

a(n) = fibonacci(n-kronecker(n,5)) % n^2 \\ Jeppe Stig Nielsen, Jul 22 2014

A298945 a(n) = F_{c-(5/c)} mod c^2, where c is the n-th composite number, F_i = A000045(i) and (5/c) is the Kronecker symbol.

Original entry on oeis.org

2, 5, 34, 21, 55, 89, 37, 160, 98, 293, 365, 150, 101, 433, 25, 665, 696, 709, 440, 994, 883, 1090, 765, 1241, 230, 1511, 1355, 257, 805, 20, 1382, 289, 2275, 1525, 1414, 821, 1373, 1820, 685, 1504, 2177, 720, 3102, 1302, 1250, 190, 2425, 2178, 2832, 3935

Offset: 1

Felix Fröhlich, Jan 30 2018

nonn

Composites c where a(n) = 0 could be called "Wall-Sun-Sun pseudoprimes" or "Fibonacci-Wieferich pseudoprimes". Do any such composites exist?

Any such c would have to be a term of A241505.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A241505, A298944, A298946.

N:= 100: # to get a(1)..a(N)
count:= 0: R:= NULL:
for n from 4 while count < N do
if not isprime(n) then
  count:= count+1;
  R:= R, combinat:-fibonacci(n - numtheory:-jacobi(5,n)) mod n^2;
fi
od:
R; # Robert Israel, Feb 02 2018

composite[n_Integer] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1] ; Array[With[{c = composite@ #}, Mod[Fibonacci[c - KroneckerSymbol[5, c]], c^2]] &, 50] (* Michael De Vlieger, Jan 31 2018, composite function by Robert G. Wilson v at A066277 *)

forcomposite(c=1, 200, print1(lift(Mod(fibonacci(c-kronecker(5, c)), c^2)), ", "))

cp's OEIS Frontend

A113649 Fibonacci(n-J(n,5)) mod n^2, where J is the Jacobi symbol.

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A298945 a(n) = F_{c-(5/c)} mod c^2, where c is the n-th composite number, F_i = A000045(i) and (5/c) is the Kronecker symbol.

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