A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Author

Gus Wiseman, Oct 31 2024

Keywords

• nonn

Comments

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

Examples

			Between prime(4) = 7 and prime(5) = 11 we have perfect-powers 8 and 9, so a(4) = 2.
		

Crossrefs

For prime-powers instead of perfect-powers we have A080101.

Non-perfect-powers in the same range are counted by A377433.

Positions of 1 are A377434.

Positions of 0 are A377436.

Positions of terms > 1 are A377466.

For powers of 2 instead of primes we have A377467, for prime-powers A244508.

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.

A046933 counts the interval from A008864(n) to A006093(n+1).

A081676 gives the greatest perfect-power <= n.

A246655 lists the prime-powers not including 1, complement A361102.

A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.

A377468 gives the least perfect-power > n.

Cf. A000015, A002808, A024619, A031218, A053707, A064113, A065514, A065890, A080769, A377051, A377282.

Programs

Mathematica

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

Formula

a(n) + A377433(n) = A046933(n) = prime(n+1) - prime(n) - 1.