A378253 Perfect powers p such that there are no other perfect powers between p and the least prime > p.

1, 4, 9, 16, 27, 36, 49, 64, 81, 100, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849

Offset: 1

Author

Gus Wiseman, Nov 26 2024

Keywords

• nonn

Comments

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

Each term is the greatest perfect power < prime(k) for some k.

Examples

			The first number line below shows the perfect powers. The second shows each prime. To get a(n), we take the last perfect power in each interval between consecutive primes, omitting the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==
		

Crossrefs

Union of A378035, restriction of A081676 to the primes.

The opposite is A378250, union of A378249 (run-lengths A378251).

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.

A377283 ranks perfect powers between primes, differences A378356.

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

Cf. A000015, A000961, A052410, A067871, A076412, A131605, A188951, A216765, A345531, A377057, A377468, A378355.

Programs

Mathematica

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