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

15, 20, 22, 295, 1257

Offset: 1

Author

Gus Wiseman, Dec 17 2024

Keywords

• nonn
• more

Comments

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

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

Examples

			The 15th and 16th perfect powers are 125 and 128, and 127 is the only prime between them, so 15 is in the sequence.
		

Crossrefs

These are the positions of 1 in A080769.

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

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

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

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

For prime powers instead of perfect powers we have A379155.

A000040 lists the primes, differences A001223.

A001597 lists the perfect powers, differences A053289.

A007916 lists the non perfect powers, differences A375706.

A069623 counts perfect powers <= n.

A076411 counts perfect powers < n.

A081676 gives the greatest perfect power <= n.

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

A377468 gives the least perfect power > n.

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

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
v=Select[Range[1000],perpowQ];
Select[Range[Length[v]-1],Length[Select[Range[v[[#]],v[[#+1]]],PrimeQ]]==1&]

Formula

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