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A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

%I A375712 #6 Sep 10 2024 08:04:33
%S A375712 1,4,7,11,12,13,14,22,25,26,29,32,35,39,40,41,42,50,53,54,61,64,70,71,
%T A375712 72,75,78,81,82,83,84,87,90,98,99,102,109,110,117,120,123,124,127,135,
%U A375712 139,140,144,151,154,155,156,157,160,163,168,169,170,173,176,179
%N A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.
%C A375712 The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
%C A375712 - A375709 (difference 1)
%C A375712 - A375710 (difference 2)
%C A375712 - A375711 (difference 3)
%C A375712 - A375712 (difference 4)
%F A375712 Complement of A375709 U A375710 U A375711.
%e A375712 The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 4 after the first, fourth, and seventh terms.
%t A375712 Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],4]
%Y A375712 For prime numbers we have A029709.
%Y A375712 Positions of 4's in A078147.
%Y A375712 A005117 lists the squarefree numbers, first differences A076259.
%Y A375712 A013929 lists the nonsquarefree numbers, first differences A078147.
%Y A375712 A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
%Y A375712 A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
%Y A375712 Cf. A007674, A014689, A029707, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A375927.
%K A375712 nonn
%O A375712 1,2
%A A375712 _Gus Wiseman_, Sep 09 2024