A112312 Least index k such that the n-th prime divides the k-th tribonacci number.

4, 8, 15, 6, 9, 7, 29, 19, 30, 78, 15, 20, 36, 83, 30, 34, 65, 69, 101, 133, 32, 19, 271, 110, 20, 187, 14, 185, 106, 173, 587, 80, 12, 35, 11, 224, 72, 38, 42, 315, 101, 26, 73, 172, 383, 27, 84, 362, 35, 250, 37, 29, 507, 305, 55, 38, 178, 332, 62, 537, 778, 459, 31, 124

Offset

1,1

Comments

The tribonacci numbers are indexed so that trib(0) = trib(1) = 0, trib(2) = 1, for n>2: trib(n) = trib(n-1) + trib(n-2) + trib(n-3). See A112618 for another version.

Brenner proves that every prime divides some tribonacci number T(n). For the similar 3-step Lucas sequence A001644, there are primes (A106299) that do not divide any term.

Links

Table of n, a(n) for n=1..64.

J. L. Brenner, Linear Recurrence Relations, Amer. Math. Monthly, Vol. 61 (1954), 171-173.

Eric Weisstein's World of Mathematics, MathWorld: Tribonacci Number

Formula

a(n) = minimum k such that prime(n) | A000073(k) and A000073(k) >= prime(n). a(n) = minimum k such that A000040(n) | A000073(k) and A000073(k) >= A000040(n).

Example

a(1) = 4 because prime(1) = 2 and tribonacci( 4) = 2.
a(2) = 8 because prime(2) = 3 and tribonacci( 8) = 24 = 3 * 2^3.
a(3) = 15 because prime(3) = 5 and tribonacci(15) = 1705 = 5 *(11 * 31).
a(4) = 6 because prime(4) = 7 and tribonacci( 6) = 7.
a(5) = 9 because prime(5) = 11 and tribonacci( 9) = 44 = 11 * 4.
a(6) = 7 because prime(6) = 13 and tribonacci( 7) = 13.
a(7) = 29 because prime(7) = 17 and tribonacci(29) = 8646064 = 17 *(2^4 * 7 * 19 * 239).

Mathematica

a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3]; f[n_] := Module[{k = 2, p = Prime[n]}, While[Mod[a[k], p] != 0, k++ ]; k]; Array[f, 64] (* Robert G. Wilson v *)

Crossrefs

Cf. A000040, A000045, A000073, A000204, A001644, A053028, A106299, A112312.

Cf. also A112618 = this sequence minus 1.

Sequence in context: A174554 A360527 A272048 * A076343 A335382 A272346

Adjacent sequences: A112309 A112310 A112311 * A112313 A112314 A112315

Keyword

easy,nonn

Author

Jonathan Vos Post, Nov 29 2005

Extensions

Corrected and extended by Robert G. Wilson v, Dec 01 2005

Status

approved