A373678 Sums of maximal runs of non-prime-powers.

1, 6, 10, 12, 29, 18, 63, 24, 26, 28, 30, 138, 117, 42, 135, 48, 153, 280, 60, 125, 131, 207, 72, 380, 80, 82, 430, 651, 297, 102, 315, 108, 333, 819, 369, 126, 259, 670, 138, 1296, 150, 770, 800, 495, 168, 513, 880, 180, 1674, 192, 585, 198, 2255, 2387, 675

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 A361102) is an interval of positions at which consecutive terms differ by one.

Examples

			The maximal runs of non-powers of primes 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

A000040 lists the primes, differences A001223.

A000961 lists all powers of primes (A246655 if not including 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).

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

Prime-power runs: A373675, min A373673, max A373674, length A174965.

Non-prime-power runs: A373678, min A373676, max A373677, length A110969.

Prime-power antiruns: A373576, min A120430, max A006549, length A373671.

Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.

Cf. A005117, A014963, A038664, A046933, A054265, A065855, A071148, A293697, A371201, A373400.

Programs

Mathematica

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