A373676 First element of each maximal run of non-prime-powers.

1, 6, 10, 12, 14, 18, 20, 24, 26, 28, 30, 33, 38, 42, 44, 48, 50, 54, 60, 62, 65, 68, 72, 74, 80, 82, 84, 90, 98, 102, 104, 108, 110, 114, 122, 126, 129, 132, 138, 140, 150, 152, 158, 164, 168, 170, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234

Offset: 1

Author

Gus Wiseman, Jun 16 2024

Keywords

• nonn

Comments

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.

The last element of the same run is A373677.

Consists of 1 and all non-prime-powers k such that k-1 is a power of a prime.

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
		

Links

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

Crossrefs

See link for prime, composite, squarefree, and nonsquarefree runs/antiruns.

For runs of powers of primes:

- length A174965

- min A373673

- max A373674

- sum A373675

For runs of non-prime-powers:

- length A110969 (firsts A373669, sorted A373670)

- min A373676 (this sequence)

- max A373677

- sum A373678

For antiruns of prime-powers:

- length A373671

- min A120430

- max A006549

- sum A373576

For antiruns of non-prime-powers:

- length A373672

- min A373575

- max A255346

- sum A373679

A000961 lists all powers of primes. A246655 is just prime-powers so lacks 1.

A025528 counts prime-powers up to n.

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

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

Cf. A000040, A007674, A014963, A038664, A053806, A054265, A068781, A072284, A073051, A373400.

Programs

Mathematica

Select[Range[100],#==1||!PrimePowerQ[#]&&PrimePowerQ[#-1]&]