A373575 Numbers k such that k and k-1 both have at least two distinct prime factors. First element of the n-th maximal antirun of non-prime-powers.

1, 15, 21, 22, 34, 35, 36, 39, 40, 45, 46, 51, 52, 55, 56, 57, 58, 63, 66, 69, 70, 75, 76, 77, 78, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 105, 106, 111, 112, 115, 116, 117, 118, 119, 120, 123, 124, 130, 133, 134, 135, 136, 141, 142, 143, 144, 145

Offset: 1

Author

Gus Wiseman, Jun 18 2024

Keywords

• nonn

Comments

The last element of the same antirun is given by A255346.

An antirun of a sequence (in this case A361102) 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

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Runs of prime-powers:

- length A174965

- min A373673

- max A373674

- sum A373675

Runs of non-prime-powers:

- length A110969

- min A373676

- max A373677

- sum A373678

Antiruns of prime-powers:

- length A373671

- min A120430

- max A006549

- sum A373576

Antiruns of non-prime-powers:

- length A373672

- min A373575 (this sequence)

- max A255346

- sum A373679

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).

Various run-lengths: A053797, A120992, A175632, A176246.

Various antirun-lengths: A027833, A373127, A373403, A373409.

Cf. A001359, A008864, A014963, A067774, A251092, A373669, A373670.

Programs

Mathematica

Select[Range[100],!PrimePowerQ[#]&&!PrimePowerQ[#-1]&]
Join[{1},SequencePosition[Table[If[PrimeNu[n]>1,1,0],{n,150}],{1,1}][[;;,2]]] (* Harvey P. Dale, Feb 23 2025 *)