A106310 - OEIS

A106310

Primes p such that p^2 divides some T(k), yet p does not divide any T(j) for any j<k, where T(n) is the n-th tribonacci number (A000073).

%I #4 Mar 30 2012 17:22:36

%S 47,617,2693

%N Primes p such that p^2 divides some T(k), yet p does not divide any T(j) for any j<k, where T(n) is the n-th tribonacci number (A000073).

%C No other p < 10^6. For Fibonacci numbers, A000045, there are no known primes with this property.

%e 47 is here because the 29th tribonacci number, 15902591, is the first tribonacci number divisible by 47 and 47^2 also divides it. Similarly, 617^2 divides T(409) and 2693^2 divides T(10553).

%t FibonacciZero[n_, kMax_, m_] := Module[{a, s, k}, a=Join[{1}, Table[0, {n-1}]]; a=Mod[a, m]; k=0; While[k++; s=Mod[Plus@@a, m]; a=RotateLeft[a]; a[[n]]=s; s>0&&k<kMax]; If[s==0, k, -1]]; Do[p=Prime[n]; zero=FibonacciZero[3, Infinity, p]; If[zero==FibonacciZero[3, zero, p^2], Print[{p, zero}]], {n, 1000}]

%K bref,hard,more,nonn

%O 1,1

%A _T. D. Noe_, May 17 2005