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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).
6
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.
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 * A373986 A136179 A185176
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Nov 16 2000
STATUS
approved