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A112269 Least index k such that the n-th prime properly divides the k-th tribonacci number.

%I A112269 #12 Mar 17 2015 23:02:21
%S A112269 5,8,15,13,9,19,29,19,30,78,15,20,36,83,30,34,65,69,101,133,32,19,271,
%T A112269 110,20,187,14,185,106,173,587,80,12,35,46,224,72,38,42,315,101,26,73,
%U A112269 172,383,27,84,362,35,250,37,29,507,305,55,38,178,332,62,537,778,459,31
%N A112269 Least index k such that the n-th prime properly divides the k-th tribonacci number.
%C A112269 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). "Properly divides" means that this sequence is "Least index k such that the n-th prime divides the k-th tribonacci number not itself the n-th prime".
%C A112269 Since the tribonacci sequence is periodic mod p for any prime p, the sequence is well-defined. - _T. D. Noe_, Dec 01 2005
%F A112269 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).
%e A112269 a(1) = 5 because prime(1) = 2 and, although tribonacci(4) = 2, the first tribonacci number properly divided by 2 is tribonacci(5) = 4.
%e A112269 a(2) = 8 because prime(2) = 3 and tribonacci(8) = 24 = 2^3 * 3.
%e A112269 a(3) = 15 because prime(3) = 5 and tribonacci(15) = 1705 = 5 * 11 * 31.
%e A112269 a(4) = 13 because prime(4) = 7 and tribonacci(13) = 504 = 2^3 * 3^2 * 7.
%e A112269 a(5) = 9 because prime(5) = 11 and tribonacci(9) = 44 = 2^2 * 11.
%e A112269 a(6) = 19 because prime(6) = 13 and tribonacci(19) = 19513 = 13 * 19 * 79.
%e A112269 a(7) = 29 because prime(7) = 17 and tribonacci(29) = 646064 = 2^4 * 7 * 17 * 19 * 239.
%t A112269 a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3]; f[n_] := Block[{k = 1, p = Prime[n]}, While[ Mod[a[k], p] != 0 || p >= a[k], k++ ]; k]; Array[f, 63] (* _Robert G. Wilson v_ *)
%Y A112269 Cf. A000040, A000045, A000073, A000204, A001644, A053028, A106299, A112312.
%K A112269 easy,nonn
%O A112269 1,1
%A A112269 _Jonathan Vos Post_, Nov 29 2005
%E A112269 Corrected and extended by _Robert G. Wilson v_, Nov 30 2005