A373672 Length of the n-th maximal antirun of non-prime-powers.

5, 3, 1, 6, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Author

Gus Wiseman, Jun 14 2024

Keywords

• nonn

Comments

An antirun of a sequence (in this case A361102 or A024619 with 1) is an interval of positions at which consecutive terms differ by more than one.

Examples

			The maximal antiruns 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
		

Links

• Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Crossrefs

For prime antiruns we have A027833.

For nonsquarefree runs we have A053797, firsts A373199.

For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.

For squarefree runs we have A120992.

For prime-power runs we have A174965.

For prime runs we have A175632.

For composite runs we have A176246, firsts A073051, sorted A373400.

For squarefree antiruns we have A373127, firsts A373128.

For composite antiruns we have A373403.

For antiruns of prime-powers:

- length A373671

- min A120430

- max A006549

For antiruns of non-prime-powers:

- length A373672 (this sequence), firsts (3,7,2,25,1,4)

- min A373575

- max A255346

A000961 lists all powers of primes. A246655 lists just prime-powers.

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

A356068 counts non-prime-powers up to n.

A361102 lists all non-prime-powers (A024619 if not including 1).

Cf. A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

Programs

Mathematica

Length/@Split[Select[Range[100],!PrimePowerQ[#]&],#1+1!=#2&]//Most

Formula

Partial sums are A356068(A255346(n)).