OFFSET

1,2

COMMENTS

A Lucas number L(n) has for multiples L(m) those for which m = n mod 2n.

However, there are composite Lucas numbers with prime indices in the sequence of Lucas numbers; these are coprime to smaller Lucas numbers (such as L(23) and L(29), which have for least prime factor 139 and 59, respectively.

L(m)|L(n) if and only if n = (2k - 1)m, with m > 1 and k > 0 (this is Theorem 16.6 in Koshy's book).

REFERENCES

Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York: John Wiley & Sons Inc. (2001) p. 200

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

MATHEMATICA

Select[LucasL[Range[60]], Not[PrimeQ[#]]&]

PROG

(Magma) [Lucas(n): n in [1..60] | not IsPrime(Lucas(n)) ]; // G. C. Greubel, Apr 21 2022

(SageMath) [lucas_number2(n, 1, -1) for n in (1..60) if not is_prime(lucas_number2(n, 1, -1))] # G. C. Greubel, Apr 21 2022

CROSSREFS

KEYWORD

nonn

AUTHOR

Alonso del Arte, Nov 19 2010

EXTENSIONS

Terms a(27) onward added by G. C. Greubel, Apr 21 2022

STATUS

approved