A373824 Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

1, 2, 11, 13, 29, 33, 45, 51, 57, 59, 69, 75, 105, 129, 211, 227, 301, 313, 321, 341, 407, 413, 447, 459, 537, 679, 709, 767, 1113, 1301, 1405, 1411, 1429, 1439, 1709, 1829, 1923, 2491, 2543, 2791, 2865, 3301, 3471, 3641, 4199, 4611, 5181, 5231, 6345, 6555

Offset

1,2

Comments

Sorted positions of first appearances in A373819.

Links

Table of n, a(n) for n=1..50.

Example

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 sorted positions of first appearances a(n).

Mathematica

t=Length/@Split[Length/@Split[Select[Range[3, 10000], PrimeQ], #1+2==#2&]];
Select[Range[Length[t]], FreeQ[Take[t, #-1], t[[#]]]&]

Crossrefs

Sorted firsts of A373819 (run-lengths of A251092).

The unsorted version is A373825.

For antiruns we have A373826, unsorted A373827.

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.

A373820 gives run-lengths of antirun-lengths, run-lengths of A027833.

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

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

Cf. A006560, A006562, A029707, A037201, A038664, A069010, A122535, A176246, A373401, A373402.

Sequence in context: A063587 A038972 A242776 * A105887 A019407 A075781

Adjacent sequences: A373821 A373822 A373823 * A373825 A373826 A373827

Keyword

nonn

Author

Gus Wiseman, Jun 21 2024

Status

approved