A373574 Numbers k such that the k-th maximal antirun of nonsquarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373409.

1, 2, 4, 6, 8, 10, 18, 52, 678

Offset: 1

Author

Gus Wiseman, Jun 10 2024

Keywords

• nonn
• more

Comments

The unsorted version is A373573.

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

Is this sequence finite? Are there only 9 terms?

Examples

			The maximal antiruns of nonsquarefree numbers begin:
   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 a(n)-th rows are:
     4    8
     9   12   16   18   20   24
    28   32   36   40   44
    49
    64   68   72   75
    81   84   88   90   92   96   98
   148  150  152
   477  480  484  486  488  490  492  495
  6345 6348 6350 6352 6354 6356 6358 6360 6363
		

Links

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

Crossrefs

For squarefree runs we have the triple (1,3,5), firsts of A120992.

For prime runs we have the triple (1,2,3), firsts of A175632.

For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.

For squarefree antiruns: A373200, firsts of A373127, unsorted A373128.

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

For prime antiruns we have A373402, firsts of A027833, unsorted A373401.

For composite antiruns we have the triple (1,2,7), firsts of A373403.

Sorted positions of first appearances in A373409.

The unsorted version is A373573.

A005117 lists the squarefree numbers, first differences A076259.

A013929 lists the nonsquarefree numbers, first differences A078147.

Cf. A007674, A025157, A049094, A061399, A068781, A072284, A077643, A110969, A251092, A294242, A373410, A373412.

Programs

Mathematica

t=Length/@Split[Select[Range[100000],!SquareFreeQ[#]&],#1+1!=#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]