A106310 Primes p such that p^2 divides tribonacci number A000073(k) for some k > 1, yet p does not divide A000073(j) for any 1 < j < k.

47, 617, 2693

Offset

1,1

Comments

No other p < 10^6.

For Fibonacci numbers (A000045), primes with a similar property are called Wall-Sun-Sun primes, but none of them are known.

Links

Table of n, a(n) for n=1..3.

Wikipedia, Wall-Sun-Sun prime.

Example

47 is here because the 30th tribonacci number, 15902591, is the first tribonacci number divisible by 47 and 47^2 also divides it.
Similarly, 617^2 divides A000073(410) and 2693^2 divides A000073(10554).

Mathematica

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}]

Crossrefs

Sequence in context: A142253 A142577 A098226 * A163709 A244880 A101793

Adjacent sequences: A106307 A106308 A106309 * A106311 A106312 A106313

Keyword

bref,hard,more,nonn

Author

T. D. Noe, May 17 2005

Extensions

Edited by Max Alekseyev, Nov 18 2025

Status

approved