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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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%I #14 Feb 03 2018 09:56:31

%S 2,5,34,21,55,89,37,160,98,293,365,150,101,433,25,665,696,709,440,994,

%T 883,1090,765,1241,230,1511,1355,257,805,20,1382,289,2275,1525,1414,

%U 821,1373,1820,685,1504,2177,720,3102,1302,1250,190,2425,2178,2832,3935

%N 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.

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

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

%H Robert Israel, <a href="/A298945/b298945.txt">Table of n, a(n) for n = 1..10000</a>

%p N:= 100: # to get a(1)..a(N)

%p count:= 0: R:= NULL:

%p for n from 4 while count < N do

%p if not isprime(n) then

%p count:= count+1;

%p R:= R, combinat:-fibonacci(n - numtheory:-jacobi(5,n)) mod n^2;

%p fi

%p od:

%p R; # _Robert Israel_, Feb 02 2018

%t 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 *)

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

%Y Cf. A000045, A241505, A298944, A298946.

%K nonn

%O 1,1

%A _Felix Fröhlich_, Jan 30 2018