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A223486 Lucas entry points: a(n) = least k such that n divides Lucas number L_k (=A000032(k), for k >= 0), or -1 if there is no such k. 3
0, 0, 2, 3, -1, 6, 4, -1, 6, -1, 5, -1, -1, 12, -1, -1, -1, 6, 9, -1, -1, 15, 12, -1, -1, -1, 18, -1, 7, -1, 15, -1, -1, -1, -1, -1, -1, 9, -1, -1, 10, -1, 22, 15, -1, 12, 8, -1, 28, -1, -1, -1, -1, 18, -1, -1, -1, 21, 29, -1, -1, 15, -1, -1, -1, -1, 34, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

If one takes L_k, for k >= 1, that is A000204, then a(1) = 1 and a(2) = 3 followed by the given numbers. This fits then with A106291(n) = A253808(n)*a(n), n >= 1 (where in A253808 a negative entry at position n indicates, as in the present sequence, that the Lucas numbers are not divisible by n. For odd primes not dividing any Lucas numbers see A053028. No power 2^m, m >= 3 divides any Lucas number, see, e.g., Vajda, p. 81). - Wolfdieter Lang, Jan 20 2015

REFERENCES

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 25.

S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

EXAMPLE

a(9) = 6 because L_6 = 18 is the first number in the Lucas sequence (A000032) that 9 divides.

MATHEMATICA

test[n_] := Module[{a, b, t, cnt = 1}, {a, b} = {2, 1}; While[cnt++; t = b; b = Mod[a + b, n]; a = t; ! (b == 0 || {a, b} == {2, 1})]; If[b == 0, cnt, -1]]; Join[{0, 0}, Table[test[i], {i, Range[3, 100]}]] (* T. D. Noe, Mar 22 2013 *)

CROSSREFS

Cf. A000032, A000204, A001177, A194363, A053028 (primes not dividing any Lucas numbers), A106291, A253808.

Sequence in context: A319192 A114576 A198724 * A263294 A205112 A173161

Adjacent sequences:  A223483 A223484 A223485 * A223487 A223488 A223489

KEYWORD

sign

AUTHOR

Casey Mongoven, Mar 20 2013

EXTENSIONS

Edited. Added "k >= 0" in the name and added cross references. - Wolfdieter Lang, Jan 20 2015

STATUS

approved

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Last modified January 16 15:53 EST 2021. Contains 340206 sequences. (Running on oeis4.)