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

1,2

Comments

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

Links

Table of n, a(n) for n=1..59.

Example

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==

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&]

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.

Sequence in context: A168434 A300416 A353134 * A038107 A377057 A303331

Adjacent sequences: A377280 A377281 A377282 * A377284 A377285 A377286

Keyword

nonn

Author

Gus Wiseman, Nov 21 2024

Status

approved