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

0,1

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

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}]

Prog

(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

Crossrefs

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

Cf. A001578 (analog for Fibonacci).

Sequence in context: A319916 A213074 A140966 * A136179 A185176 A198315

Adjacent sequences: A058033 A058034 A058035 * A058037 A058038 A058039

Keyword

nonn

Author

Robert G. Wilson v, Nov 16 2000

Status

approved