A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

1, 5, 7, 12, 18, 190, 28, 109, 40, 28195574, 53

Offset: 1

Author

Gus Wiseman, Jun 14 2024

Keywords

• nonn
• more

Comments

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.

Are there only 9 terms?

From David A. Corneth, Jun 14 2024: (Start)

No. a(10) exists.

Between the prime 144115188075855859 and 144115188075855872 = 2^57 there are 12 non-prime-powers so a(12) exists. (End)

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

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

For squarefree runs we have firsts of A120992.

For prime-powers runs we have firsts of A174965.

For prime runs we have firsts of A251092 or A175632.

For squarefree antiruns we have A373128, firsts of A373127.

For nonsquarefree runs we have A373199, firsts of A053797.

The sorted version is A373670.

For antiruns we have firsts of A373672.

For runs of non-prime-powers:

- length A110969

- min A373676

- max A373677

- sum A373678

A000961 lists the powers of primes (including 1).

A005117 lists the squarefree numbers, first differences A076259.

A013929 lists the nonsquarefree numbers, first differences A078147.

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

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

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

Cf. A007053, A008864, A014963, A027833, A038664, A054265, A067774, A356068, A373401, A373403, A373671.

Programs

Mathematica

q=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#1]]&];
Table[Position[q,k][[1,1]],{k,spna[q]}]