A378617 First differences of A378249 (next perfect power after prime(n)).

0, 4, 0, 8, 0, 9, 0, 0, 7, 0, 17, 0, 0, 0, 15, 0, 0, 17, 0, 0, 0, 19, 0, 0, 21, 0, 0, 0, 0, 7, 16, 0, 0, 25, 0, 0, 0, 0, 27, 0, 0, 0, 0, 20, 0, 0, 9, 18, 0, 0, 0, 0, 13, 33, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 19, 0, 18, 0, 0, 0, 39, 0, 0, 0, 0, 0, 41, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Dec 09 2024

Keywords

• nonn

Comments

This is the next perfect power after prime(n+1), minus the next perfect power after prime(n).

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

Crossrefs

Positions of positives are A377283.

Positions of zeros are A377436.

The restriction to primes has first differences A377468.

A version for nonsquarefree numbers is A377784, differences of A377783.

The opposite is differences of A378035 (restriction of A081676).

First differences of A378249, run-lengths A378251.

Without zeros we have differences of A378250.

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.

A377432 counts perfect powers between primes.

A378356 - 1 gives next prime after perfect powers, union A378365 - 1.

Cf. A000015, A007918, A023055, A045542, A052410, A065514, A076412, A188951, A216765, A377431, A377434.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,Prime[n],Not@*perpowQ],{n,100}]//Differences