A058036 Smallest primitive prime factor of the n-th Lucas number (A000032); i.e., L(n), L(0) = 2, L(1) = 1 and L(n) = L(n-1) + L(n-2).

2, 1, 3, 1, 7, 11, 1, 29, 47, 19, 41, 199, 23, 521, 281, 31, 2207, 3571, 107, 9349, 2161, 211, 43, 139, 1103, 101, 90481, 5779, 14503, 59, 2521, 3010349, 1087, 9901, 67, 71, 103681, 54018521, 29134601, 79, 1601, 370248451, 83, 6709, 263, 181, 4969

Offset: 0

Author

Robert G. Wilson v, Nov 16 2000

Keywords

• nonn

Comments

A Lucas number can have more than one primitive factor; the primitive factors of L(22) are 43 and 307.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000 (using Blair Kelly's data).
• Mansur S. Boase, A Result About the Primes Dividing Fibonacci Numbers, The Fibonacci Quarterly, 39.5 (2001) 386.
• J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260, S1-S15. Math. Rev. 89h:11002.
• Blair Kelly, Fibonacci and Lucas Factorizations

Crossrefs

Cf. A000032, A086600 (number of primitive prime factors in L(n)).

Cf. A001578 (analog for Fibonacci).

Programs

Mathematica

a=3; b=-1; prms={}; Table[c=a+b; a=b; b=c; f=First/@FactorInteger[c]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {47}]

PARI

lucas(n) = fibonacci(n+1)+fibonacci(n-1); \\ A000032
a(n) = {n++; my(v = vector(n, k, k--; lucas(k))); my(vf = vector(n, k, factor(v[k])[,1]~)); for (k=1, n-1, vf[n] = setminus(vf[n], vf[k]);); if (#vf[n], vecmin(vf[n]), 1);} \\ Michel Marcus, May 11 2021