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.

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

Author

Felix Fröhlich, Jan 30 2018

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

Cf. A000045, A241505, A298944, A298946.

Programs

Maple

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

Mathematica

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 *)

PARI

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