A378368 - OEIS

%I #8 Dec 20 2024 02:07:56

%S 15,20,22,295,1257

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

%C Perfect powers (A001597) are 1 and numbers with a proper integer root.

%C The perfect powers themselves are given by A001597(a(n)) = A378355(n).

%F We have A001597(a(n)) = A378355(n) < A178700(n) < A378374(n).

%e 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 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;

%t v=Select[Range[1000],perpowQ];

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

%Y These are the positions of 1 in A080769.

%Y The next prime after A001597(a(n)) is A178700(n).

%Y For no (instead of one) perfect powers we have A274605.

%Y Swapping 'prime' and 'perfect power' gives A377434, unique case of A377283.

%Y The next perfect power after A001597(a(n)) is A378374(n).

%Y For prime powers instead of perfect powers we have A379155.

%Y A000040 lists the primes, differences A001223.

%Y A001597 lists the perfect powers, differences A053289.

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

%Y A069623 counts perfect powers <= n.

%Y A076411 counts perfect powers < n.

%Y A081676 gives the greatest perfect power <= n.

%Y A377432 counts perfect powers between primes, see A377436, A377466.

%Y A377468 gives the least perfect power > n.

%Y Cf. A023055, A045542, A046933, A052410, A053607, A067871, A076412, A080101, A375740, A377287.

%K nonn,more,new

%O 1,1

%A _Gus Wiseman_, Dec 17 2024