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Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.
5

%I #6 Sep 10 2024 08:04:42

%S 5,6,9,19,20,21,33,34,36,49,57,58,62,63,66,76,77,88,89,91,96,97,103,

%T 104,113,114,118,119,130,131,132,136,142,149,150,161,162,174,175,187,

%U 188,189,190,201,202,206,215,217,218,225,226,231,232,245,246,249,253

%N Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

%C The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:

%C - A375709 (difference 1)

%C - A375710 (difference 2)

%C - A375711 (difference 3)

%C - A375712 (difference 4)

%F Complement of A375709 U A375711 U A375712.

%e The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 2 after the fifth and sixth terms.

%t Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]],2]

%Y Positions of 2's in A078147.

%Y For prime numbers we have A029707.

%Y For nonprime numbers we appear to have A014689.

%Y A005117 lists the squarefree numbers, first differences A076259.

%Y A013929 lists the nonsquarefree numbers, first differences A078147.

%Y A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.

%Y A375707 counts squarefree numbers between consecutive nonsquarefree numbers.

%Y Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A373573, A375927.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 09 2024