A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

1, 5, 7, 12, 18, 28, 40, 53, 71, 109, 170, 190, 198, 207, 236, 303, 394, 457, 606, 774, 1069, 1100, 1225, 1881, 1930, 1952, 2247, 2281, 3140, 3368, 3451, 3493, 3713, 3862, 4595, 4685, 6625, 8063, 8121, 8783, 12359, 12650, 14471, 14979, 15901, 17129, 19155

Offset: 1

Author

Gus Wiseman, Jun 15 2024

Keywords

• nonn

Comments

The unsorted version is A373669.

Examples

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60
So the a(n)-th runs begin:
   1
  14  15
  20  21  22
  33  34  35  36
  54  55  56  57  58
		

Crossrefs

For nonsquarefree runs we have A373199 (if increasing), firsts of A053797.

For squarefree antiruns see A373200, unsorted A373128, firsts of A373127.

For composite runs we have A373400, unsorted A073051, firsts of A176246.

For prime antiruns we have A373402.

For runs of non-prime-powers:

- length A110969, firsts A373669, sorted A373670 (this sequence):

- min A373676

- max A373677

- sum A373678

For runs of prime-powers:

- length A174965

- min A373673

- max A373674

- sum A373675

A000961 lists the powers of primes (including 1).

A057820 gives first differences of consecutive prime-powers, gaps A093555.

A361102 lists the non-prime-powers, without 1 A024619.

Cf. A005381, A008864, A014963, A027833, A038664, A356068, A373401, A373574.

Programs

Mathematica

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