A172159 Nonprime Lucas numbers.

1, 4, 18, 76, 123, 322, 843, 1364, 5778, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 4870847, 7881196, 12752043, 20633239, 33385282, 87403803, 141422324, 228826127, 599074578, 969323029, 1568397607, 2537720636, 4106118243, 10749957122, 17393796001, 28143753123

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

Cf. A000032 (Lucas numbers), A002808 (composite numbers).

Sequence in context: A245127 A037674 A066259 * A084213 A048664 A108012

Adjacent sequences: A172156 A172157 A172158 * A172160 A172161 A172162

Keyword

nonn

Author

Alonso del Arte, Nov 19 2010

Extensions

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

Status

approved