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

1,1

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.

Links

Table of n, a(n) for n=1..87.

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

Example

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}

Mathematica

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

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.

Sequence in context: A220279 A221188 A220532 * A060295 A102912 A064850

Adjacent sequences: A373406 A373407 A373408 * A373410 A373411 A373412

Keyword

nonn

Author

Gus Wiseman, Jun 06 2024

Status

approved