A307213 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.

1, 2, 5, 9, 10, 11, 16, 17, 23, 30, 31, 39, 40, 41, 50, 51, 52, 43, 53, 64, 65, 66, 67, 79, 92, 93, 94, 95, 96, 97, 111, 112, 113, 128, 129, 130, 131, 147, 148, 149, 166, 167, 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

Offset: 1

Author

Christopher Cormier, Mar 28 2019

Keywords

• nonn

Comments

"Cycle" in the definition, means that if no prime is found, go back to the start of the set.

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.

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

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A307213

Crossrefs

Cf. A034856, A095998, A309121.

Programs

Mathematica

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 *)

PARI

See Links section.

Extensions

Edited by N. J. A. Sloane, Jul 13 2019