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%I #30 Jul 14 2019 15:23:17
%S 1,2,5,9,10,11,16,17,23,30,31,39,40,41,50,51,52,43,53,64,65,66,67,79,
%T 92,93,94,95,96,97,111,112,113,128,129,130,131,147,148,149,166,167,
%U 185,186,187,169,170,171,172,173,192,193,213,214,215,216,217,218,219,220,221,222,223,244,245,246,247,248,249,250,229
%N Divide the natural numbers into sets of successive sizes 3,4,5,6,7,...,, starting with {1,2,3}. Cycle through each set until you reach a prime; if the prime was the n-th element in its set, jump to the n-th element of the next set.
%C "Cycle" in the definition, means that if no prime is found, go back to the start of the set.
%C If a set does not contain a prime, the sequence goes into an infinite loop, but it is conjectured that this does not happen since the sets are of increasing length.
%C The sets (rather intervals) are I_j = [(j^2 + 3*j - 2)/2, j*(j + 5)/2] =[A034856(j), A095998(j)], for j >= 1. For the number of primes in these intervals see A309121. - _Wolfdieter Lang_, Jul 13 2019
%H Rémy Sigrist, <a href="/A307213/b307213.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A307213/a307213_1.gp.txt">PARI program for A307213</a>
%e The first set is {1,2,3}. We look at 1 then 2. 2 is prime, and it is the second number in the set. The next set is {4,5,6,7}. So we jump to the second element, 5. 5 is also prime, so we jump to the second element of the next set, {8,9,10,11,12}, which is 9, etc. If we reach the end of a set without reaching a prime, we loop back to the first element, which is the only way for a(n) < a(n-1) to happen.
%t Nest[Append[#1, {If[#3 <= Length@ #4, #3, #3 - Length@ #4], If[#2 == #3, {#4[[#3]]}, Join[#4, #4][[#2 ;; #3]]], #4}] & @@ {#1, #2, If[PrimeQ[#3[[#2]] ], #2, #2 + FirstPosition[RotateLeft[#3, #2], _?PrimeQ][[1]] ], #3} & @@ {#1, #2, Range[#3, #3 + #4]} & @@ {#, #[[-1, 1]], 1 + Max@ #[[-1, -1]], Length@ # + 2} &, {{#1, #2[[1 ;; #1]], #2} & @@ {FirstPosition[#, _?PrimeQ][[1]], #}} &@ Range@ 3, 19][[All, 2]] // Flatten (* _Michael De Vlieger_, Mar 31 2019 *)
%o (PARI) See Links section.
%Y Cf. A034856, A095998, A309121.
%K nonn
%O 1,2
%A _Christopher Cormier_, Mar 28 2019
%E Edited by _N. J. A. Sloane_, Jul 13 2019
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