A377283 Nonnegative integers k such that either k = 0 or there is a perfect power x in the range prime(k) < x < prime(k+1).

0, 2, 4, 6, 9, 11, 15, 18, 22, 25, 30, 31, 34, 39, 44, 47, 48, 53, 54, 61, 66, 68, 72, 78, 85, 92, 97, 99, 105, 114, 122, 129, 137, 146, 154, 162, 168, 172, 181, 191, 200, 210, 217, 219, 228, 240, 251, 263, 269, 274, 283, 295, 306, 309, 319, 327, 329, 342, 357

Offset: 1

Author

Gus Wiseman, Nov 21 2024

Keywords

• nonn

Comments

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

Examples

			The first number-line below shows the perfect powers. The second shows each positive integer k at position prime(k).
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==
		

Crossrefs

A version for prime powers is A377057, exclusive A377287.

A version for squarefree numbers is A377431.

Positions of positive terms in A377432 (counts perfect powers between primes).

The case of a unique choice is A377434 (a subset).

The complement (no choices) is A377436.

The case of at least two choices is A377466 (a subset).

Positions of last appearances in A378249.

First-differences are A378251.

This is A378365 - 1, union of A378356 - 1.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

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.

A131605 lists perfect powers that are not prime powers.

Cf. A000015, A045542, A065514, A076412, A081676, A188951, A216765, A345531, A377468, A378035, A378250.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[0,100],#==0||Length[Select[Range[Prime[#]+1,Prime[#+1]-1],perpowQ]]>0&]