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A378368 Positions (in A001597) of consecutive perfect powers with a unique prime between them.

%I A378368 #8 Dec 20 2024 02:07:56
%S A378368 15,20,22,295,1257
%N A378368 Positions (in A001597) of consecutive perfect powers with a unique prime between them.
%C A378368 Perfect powers (A001597) are 1 and numbers with a proper integer root.
%C A378368 The perfect powers themselves are given by A001597(a(n)) = A378355(n).
%F A378368 We have A001597(a(n)) = A378355(n) < A178700(n) < A378374(n).
%e A378368 The 15th and 16th perfect powers are 125 and 128, and 127 is the only prime between them, so 15 is in the sequence.
%t A378368 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A378368 v=Select[Range[1000],perpowQ];
%t A378368 Select[Range[Length[v]-1],Length[Select[Range[v[[#]],v[[#+1]]],PrimeQ]]==1&]
%Y A378368 These are the positions of 1 in A080769.
%Y A378368 The next prime after A001597(a(n)) is A178700(n).
%Y A378368 For no (instead of one) perfect powers we have A274605.
%Y A378368 Swapping 'prime' and 'perfect power' gives A377434, unique case of A377283.
%Y A378368 The next perfect power after A001597(a(n)) is A378374(n).
%Y A378368 For prime powers instead of perfect powers we have A379155.
%Y A378368 A000040 lists the primes, differences A001223.
%Y A378368 A001597 lists the perfect powers, differences A053289.
%Y A378368 A007916 lists the non perfect powers, differences A375706.
%Y A378368 A069623 counts perfect powers <= n.
%Y A378368 A076411 counts perfect powers < n.
%Y A378368 A081676 gives the greatest perfect power <= n.
%Y A378368 A377432 counts perfect powers between primes, see A377436, A377466.
%Y A378368 A377468 gives the least perfect power > n.
%Y A378368 Cf. A023055, A045542, A046933, A052410, A053607, A067871, A076412, A080101, A375740, A377287.
%K A378368 nonn,more
%O A378368 1,1
%A A378368 _Gus Wiseman_, Dec 17 2024