A373825 Position of first appearance of n in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

1, 2, 13, 11, 105, 57, 33, 69, 59, 29, 227, 129, 211, 341, 75, 321, 51, 45, 407, 313, 459, 301, 767, 1829, 413, 537, 447, 1113, 1301, 1411, 1405, 2865, 1709, 1429, 3471, 709, 2543, 5231, 1923, 679, 3301, 2791, 6555, 5181, 6345, 11475, 2491, 10633

Offset: 1

Author

Gus Wiseman, Jun 21 2024

Keywords

• nonn

Comments

Positions of first appearances in A373819.

Examples

			The runs of odd primes differing by 2 begin:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...
which have runs beginning:
  3
  2 2
  1
  2
  1
  2
  1 1
  2
  1
  2
  1 1 1 1
  2 2
  1 1 1
with lengths:
1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3, ...
with positions of first appearances a(n).
		

Crossrefs

Firsts of A373819 (run-lengths of A251092).

For antiruns we have A373827 (sorted A373826), firsts of A373820, run-lengths of A027833 (partial sums A029707, firsts A373401, sorted A373402).

The sorted version is A373824.

A000040 lists the primes.

A001223 gives differences of consecutive primes (firsts A073051), run-lengths A333254 (firsts A335406), run-lengths of run-lengths A373821.

A046933 counts composite numbers between primes.

A065855 counts composite numbers up to n.

A071148 gives partial sums of odd primes.

For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.

For composite runs: A005381, A054265, A068780, A176246, A373403, A373404.

Cf. A006560, A006562, A037201, A038664, A069010, A122535.

Programs

Mathematica

t=Length/@Split[Length/@Split[Select[Range[3,10000], PrimeQ],#1+2==#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]