A378355 Numbers appearing exactly once in A378035 (greatest perfect power < prime(n)).

125, 216, 243, 64000, 1295029, 2535525316, 542939080312

Offset: 1

Author

Gus Wiseman, Nov 26 2024

Keywords

• nonn
• more

Comments

These are perfect-powers p such that the interval from p to the next perfect power contains a unique prime.

Is this sequence infinite? See A178700.

Examples

			We have 125 because 127 is the only prime between 125 and 128.
		

Crossrefs

The next prime is A178700.

Singletons in A378035 (union A378253), restriction of A081676.

The next perfect power is A378374.

Swapping primes and perfect powers gives A379154, unique case of A377283.

A000040 lists the primes, differences A001223.

A001597 lists the perfect powers, differences A053289.

A007916 lists the not perfect powers, differences A375706.

A069623 counts perfect powers <= n.

A076411 counts perfect powers < n.

A377432 counts perfect powers between primes, see A377434, A377436, A377466.

A378249 gives least perfect power > prime(n) (run-lengths A378251), restrict of A377468.

Cf. A000961, A031218, A045542, A052410, A067871, A076412, A080769, A131605, A188951, A216765, A378250, A378356.

Programs

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
y=Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==1&]

Formula

A151800(a(n)) = A178700(n).