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A376163 Positions of adjacent non-prime-powers (inclusive, so 1 is a prime-power) differing by 1.

%I A376163 #5 Sep 13 2024 06:58:22
%S A376163 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 A376163 50,51,52,53,55,56,57,58,59,60,62,63,66,67,70,71,73,74,75,76,77,78,80,
%U A376163 81,84,86,87,88,89,92,93,94,95,96,97,98,99,102,103,104,105
%N A376163 Positions of adjacent non-prime-powers (inclusive, so 1 is a prime-power) differing by 1.
%e A376163 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 A376163 ce=Select[Range[2,100],!PrimePowerQ[#]&];
%t A376163 Select[Range[Length[ce]-1],ce[[#+1]]==ce[[#]]+1&]
%Y A376163 For prime-powers inclusive (A000961) we have A375734, differences A373671.
%Y A376163 For nonprime numbers (A002808) we have A375926, differences A373403.
%Y A376163 For prime-powers exclusive (A246655) we have A375734(n+1) + 1.
%Y A376163 First differences are A373672.
%Y A376163 The exclusive version is a(n) - 1 = A375713.
%Y A376163 Positions of 1's in A375735.
%Y A376163 For non-perfect-powers we have A375740.
%Y A376163 Prime-powers inclusive:
%Y A376163 - terms: A000961
%Y A376163 - differences: A057820
%Y A376163 - runs: A373675, A373673, A373674, A174965
%Y A376163 - antiruns: A373576, A120430, A006549, A373671
%Y A376163 Non-prime-powers inclusive:
%Y A376163 - terms: A361102
%Y A376163 - differences: A375708
%Y A376163 - runs: A373678, A373676, A373677, A110969
%Y A376163 - antiruns: A373679, A373575, A255346, A373672
%Y A376163 A000040 lists all of the primes, differences A001223.
%Y A376163 A007916 lists non-perfect-powers, differences A375706.
%Y A376163 Cf. A046933, A053289, A073783, A093555, A176246, A251092, A375714.
%K A376163 nonn
%O A376163 1,1
%A A376163 _Gus Wiseman_, Sep 13 2024