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A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

%I A375734 #18 Sep 24 2024 11:32:17
%S A375734 1,2,3,5,6,10,17,43,70,1077,6635,12369,43578,105102700
%N A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.
%C A375734 The corresponding prime-powers A246655(a(n)) are given by A006549.
%C A375734 From A006549, it is not known whether this sequence is infinite.
%F A375734 Numbers k such that A246655(k+1) - A246655(k) = 1.
%F A375734 The inclusive version is a(n) + 1 shifted.
%e A375734 The fifth prime-power is 7 and the sixth is 8, so 5 is in the sequence.
%t A375734 Join@@Position[Differences[Select[Range[100],PrimePowerQ]],1]
%Y A375734 For nonprime numbers (A002808) we have A375926, differences A373403.
%Y A375734 Positions of 1's in A057820.
%Y A375734 First differences are A373671.
%Y A375734 For nonsquarefree numbers we have A375709, differences A373409.
%Y A375734 For non-prime-powers we have A375713.
%Y A375734 For non-perfect-powers we have A375740.
%Y A375734 For squarefree numbers we have A375927, differences A373127.
%Y A375734 Prime-powers:
%Y A375734 - terms: A000961, complement A024619.
%Y A375734 - differences: A057820.
%Y A375734 - runs: A373675, A373673, A373674, A174965
%Y A375734 - anti-runs: A373576, A120430, A006549, A373671
%Y A375734 Non-prime-powers:
%Y A375734 - terms: A361102
%Y A375734 - differences: A375708
%Y A375734 - runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670).
%Y A375734 - anti-runs: A373679, A373575, A255346, A373672
%Y A375734 A000040 lists all of the primes, differences A001223.
%Y A375734 A025528 counts prime-powers up to n.
%Y A375734 Cf. A007916, A014963, A027833, A046933, A053289, A093555, A375714.
%K A375734 nonn,more
%O A375734 1,2
%A A375734 _Gus Wiseman_, Sep 04 2024
%E A375734 a(14) from _Amiram Eldar_, Sep 24 2024