A241675 a(n) = |{0 < k < n/2: k is a Fibonacci number with x^2 == k (mod n) for no integer x}|.
0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 3, 3, 4, 2, 1, 4, 4, 3, 4, 4, 3, 5, 2, 5, 4, 2, 5, 4, 4, 4, 3, 5, 2, 5, 4, 5, 6, 2, 2, 6, 4, 5, 4, 5, 5, 5, 5, 5, 6, 4, 3, 5, 3, 3, 6, 6, 6, 5, 5, 3, 5, 6, 3, 7, 4, 4, 5, 6, 7, 5, 2, 7
Offset: 1
Keywords
Examples
a(5) = 1 since x^2 == F(3) = 2 (mod 5) for no integer x, but 1^2 == F(1) = F(2) = 1 (mod 5), where F(n) denotes the n-th Fibonacci number given by A000045. a(7) = 1 since x^2 == F(4) = 3 (mod 7) for no integer x. a(22) = 2 since there is no integer x such that x^2 == F(3) = 2 (mod 22) or x^2 == F(6) = 8 (mod 22). a(23) = 1 since x^2 == F(5) = 5 (mod 23) for no integer x.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
Programs
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Mathematica
f[k_]:=Fibonacci[k] Do[m=0;Do[If[f[k]>=n/2,Goto[bb]];Do[If[Mod[i^2,n]==f[k],Goto[aa]],{i,0,n/2}];m=m+1;Label[aa];Continue,{k,2,(n+1)/2}];Label[bb];Print[n," ",m];Continue,{n,1,80}]
Comments