A378374 Perfect powers p such that the interval from the previous perfect power to p contains a unique prime.

128, 225, 256, 64009, 1295044

Offset: 1

Author

Gus Wiseman, Dec 17 2024

Keywords

• nonn
• more

Comments

Also numbers appearing exactly once in A378249.

Examples

			The consecutive perfect powers 125 and 128 have interval (125, 126, 127, 128) with unique prime 127, so 128 is in the sequence.
		

Crossrefs

The previous prime is A178700.

For prime powers instead of perfect powers we have A345531, difference A377281.

Opposite singletons in A378035 (union A378253), restriction of A081676.

For squarefree numbers we have A378082, see A377430, A061398, A377431, A068360.

Singletons in A378249 (run-lengths A378251), restriction of A377468 to the primes.

If the same interval contains at least one prime we get A378250.

For next instead of previous perfect power we have A378355.

Swapping "prime" with "perfect power" gives A378364.

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.

A080769 counts primes between perfect powers.

Cf. A023055, A045542, A052410, A216765, A376560, A377287, 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

We have a(n) < A178700(n) < A378355(n).