A373409 Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

2, 6, 2, 5, 2, 1, 6, 4, 2, 7, 1, 5, 2, 2, 1, 4, 4, 3, 6, 2, 2, 4, 7, 5, 7, 1, 1, 6, 6, 2, 3, 4, 7, 3, 3, 5, 1, 3, 1, 3, 2, 2, 3, 5, 5, 7, 1, 5, 7, 5, 1, 8, 4, 2, 5, 2, 2, 3, 3, 1, 7, 3, 4, 7, 1, 5, 2, 5, 2, 6, 7, 6, 7, 5, 1, 2, 3, 5, 6, 4, 1, 3, 5, 7, 2, 3, 2

Offset: 1

Author

Gus Wiseman, Jun 06 2024

Keywords

• nonn

Comments

An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.

Conjecture: The maximum is 9, and there is no antirun of more than 9 nonsquarefree numbers. Confirmed up to 100,000,000.

Examples

			Row-lengths of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99
The first maximal antirun of length 9 is the following, shown with prime indices:
  6345: {2,2,2,3,15}
  6348: {1,1,2,9,9}
  6350: {1,3,3,31}
  6352: {1,1,1,1,78}
  6354: {1,2,2,71}
  6356: {1,1,4,49}
  6358: {1,5,7,7}
  6360: {1,1,1,2,3,16}
  6363: {2,2,4,26}
		

Links

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

Crossrefs

Positions of first appearances are A373573, sorted A373574.

Functional neighbors: A027833, A053797, A068781, A373127, A373403, A373410, A373412.

A005117 lists the squarefree numbers, first differences A076259.

A013929 lists the nonsquarefree numbers, first differences A078147.

Cf. A025157, A049093, A061398, A061399, A077641, A077643, A110969, A174965, A294242, A350842, A373198.

Programs

Mathematica

Length/@Split[Select[Range[1000],!SquareFreeQ[#]&],#1+1!=#2&]//Most