%I #5 Sep 13 2024 06:58:22

%S 4,7,8,14,15,16,18,19,22,23,26,27,29,30,31,32,35,37,39,40,43,44,45,46,

%T 50,51,52,53,55,56,57,58,59,60,62,63,66,67,70,71,73,74,75,76,77,78,80,

%U 81,84,86,87,88,89,92,93,94,95,96,97,98,99,102,103,104,105

%N Positions of adjacent non-prime-powers (inclusive, so 1 is a prime-power) differing by 1.

%e The non-prime-powers (inclusive) are 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ... which increase by 1 after positions 4, 7, 8, ...

%t ce=Select[Range[2,100],!PrimePowerQ[#]&];

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

%Y For prime-powers inclusive (A000961) we have A375734, differences A373671.

%Y For nonprime numbers (A002808) we have A375926, differences A373403.

%Y For prime-powers exclusive (A246655) we have A375734(n+1) + 1.

%Y First differences are A373672.

%Y The exclusive version is a(n) - 1 = A375713.

%Y Positions of 1's in A375735.

%Y For non-perfect-powers we have A375740.

%Y Prime-powers inclusive:

%Y - terms: A000961

%Y - differences: A057820

%Y - runs: A373675, A373673, A373674, A174965

%Y - antiruns: A373576, A120430, A006549, A373671

%Y Non-prime-powers inclusive:

%Y - terms: A361102

%Y - differences: A375708

%Y - runs: A373678, A373676, A373677, A110969

%Y - antiruns: A373679, A373575, A255346, A373672

%Y A000040 lists all of the primes, differences A001223.

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

%Y Cf. A046933, A053289, A073783, A093555, A176246, A251092, A375714.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 13 2024