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%I #15 Dec 02 2015 12:04:38
%S 0,1,2,4,5,8,10,11,16,20,23,24,26,29,34,38,41,48,50,51,56,60,63,68,80,
%T 81,90,92,95,96,102,109,120,124,129,130,138,141,142,148,149,152,156,
%U 159,164,168,171,182,188,193,196,198,199,202,206,209,216,218,219,222,232
%N a(0) = 0; a(1) = 1; a(2) = 2; a(n) = smallest number greater than the previous term such that the sum of three successive terms is a prime.
%C Slowest increasing sequence where 3 consecutive integers sum up to a prime.
%C In a string there can be at most two consecutive integers, e.g., (10, 11). More generally, three consecutive terms cannot be in arithmetic progression.
%H Harvey P. Dale, <a href="/A073628/b073628.txt">Table of n, a(n) for n = 0..1000</a>
%e 0 + 1 + 2 = 3, which is prime; 1 + 2 + 4 = 7, which is prime; 2 + 4 + 5 = 11, which is prime.
%t n1 = 0; n2 = 1; counter = 1; maxnumber = 10^4; Do[ If[PrimeQ[n1 + n2 + n], {sol[counter] = n; counter = counter + 1; n1 = n2; n2 = n}], {n, 2, maxnumber}]; Table[sol[j], {j, 1, counter}]\) (* Ben Ross (bmr180(AT)psu.edu), Jan 29 2006 *)
%t nxt[{a_,b_,c_}]:={b,c,Module[{x=c+1},While[!PrimeQ[b+c+x],x++];x]}; Transpose[ NestList[nxt,{0,1,2},60]][[1]] (* _Harvey P. Dale_, Jun 10 2013 *)
%Y Cf. A073627.
%K nonn
%O 0,3
%A _Amarnath Murthy_, Aug 08 2002
%E More terms from _Matthew Conroy_, Sep 09 2002
%E Entry revised by _N. J. A. Sloane_, Mar 25 2007