A104622 Indices of prime values of heptanacci-Lucas numbers A104621.

0, 2, 3, 5, 7, 10, 17, 24, 25, 26, 28, 38, 40, 49, 62, 79, 89, 114, 140, 145, 182, 248, 353, 437, 654, 702, 784, 921, 931, 986, 1206, 2136, 2137, 3351, 5411, 13264, 13757, 16348, 27087, 27160

Offset

1,2

Comments

The 7th-order linear recurrence A104622 (heptanacci-Lucas numbers) is a generalization of the Lucas sequence A000032. T. D. Noe and I have noted that the heptanacci-Lucas numbers have many more primes than the corresponding heptanacci (see A104414) which he found has only the first 3 primes that I identified through the first 5000 values, whereas these heptanacci-Lucas numbers have 17 primes among the first 100 values. For semiprimes in heptanacci-Lucas numbers, see A104623.

Links

Table of n, a(n) for n=1..40.

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math.CO/0210201 v1, Oct 14 2002

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Formula

Prime values of the heptanacci-Lucas numbers, which are defined by: a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63, for n > 6: a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7).

Example

A104621(0) = 7,
A104621(2) = 3,
A104621(3) = 7,
A104621(5) = 31,
A104621(7) = 127,
A104621(10) = 983,
A104621(17) = 122401,
A104621(24) = 15231991.

Mathematica

a[0] = 7; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[5] = 31; a[6] = 63; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4] + a[n - 5] + a[n - 6] + a[n - 7]; Do[ If[ PrimeQ[ a[n]], Print[n]], {n, 5000}] (* Robert G. Wilson v, Mar 17 2005 *)
Flatten[Position[LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {7, 1, 3, 7, 15, 31, 63}, 28000], _?PrimeQ]]-1 (* Harvey P. Dale, Jan 02 2016 *)

Crossrefs

Cf. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577, A104621, A104623.

Sequence in context: A271063 A182521 A283809 * A033843 A075657 A039711

Adjacent sequences: A104619 A104620 A104621 * A104623 A104624 A104625

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 17 2005

Extensions

More terms from T. D. Noe and Robert G. Wilson v, Mar 17 2005

Status

approved