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Positions of adjacent non-prime-powers (exclusive) differing by more than 1.
2

%I #7 Sep 13 2024 06:53:06

%S 1,2,3,4,6,7,10,11,12,13,14,18,21,22,25,26,29,34,35,37,39,42,43,48,49,

%T 50,55,62,65,66,69,70,73,80,83,84,86,91,92,101,102,107,112,115,116,

%U 119,124,125,134,135,138,139,150,161,164,165,168,173,174,175,182

%N Positions of adjacent non-prime-powers (exclusive) differing by more than 1.

%F The inclusive version is a(n+1) - 1.

%e The non-prime-powers (exclusive) are 1, 6, 10, 12, 14, 15, 18, 20, ... which increase by more than 1 after positions 1, 2, 3, 4, 6, 7, ...

%t ce=Select[Range[100],!PrimePowerQ[#]&];

%t Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]

%Y For prime-powers inclusive (A000961) we have A376163, differences A373672.

%Y For nonprime numbers (A002808) we have A014689, differences A046933.

%Y First differences are A110969.

%Y The complement is A375713.

%Y For non-perfect-powers we have A375714, complement A375740.

%Y The complement for prime-powers (exclusive) is A375734, differences A373671.

%Y The complement for nonprime numbers is A375926, differences A373403.

%Y A000040 lists the prime numbers, differences A001223.

%Y A000961 lists prime-powers (inclusive), differences A057820.

%Y A007916 lists non-perfect-powers, differences A375706.

%Y A024619 lists non-prime-powers (inclusive), differences A375735.

%Y A246655 lists prime-powers (exclusive), differences A174965.

%Y A361102 lists non-prime-powers (exclusive), differences A375708.

%Y Cf. A006549, A053289, A073783, A093555, A120430, A176246, A251092.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 12 2024