A377281 Difference between the n-th prime and the next prime-power (exclusive).

1, 1, 2, 1, 2, 3, 2, 4, 2, 2, 1, 4, 2, 4, 2, 6, 2, 3, 4, 2, 6, 2, 6, 8, 4, 2, 4, 2, 4, 8, 1, 6, 2, 10, 2, 6, 6, 4, 2, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 5, 6, 6, 2, 6, 4, 2, 6, 14, 4, 2, 4, 14, 6, 6, 2, 4, 6, 2, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6

Offset: 1

Author

Gus Wiseman, Oct 23 2024

Keywords

• nonn

Examples

			The twelfth prime is 37, with next prime-power 41, so a(12) = 4.
		

Crossrefs

For prime instead of prime-power we have A001223.

For powers of two instead of primes we have A013597, A014210, A014234, A244508, A304521.

This is the restriction of A377282 to the prime numbers.

For previous instead of next prime-power we have A377289, restriction of A276781.

A000015 gives the least prime-power >= n.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820, complement A361102.

A031218 gives the greatest prime-power <= n.

A080101 counts prime-powers between primes (exclusive), cf. A377286, A377287, A377288.

A246655 lists the prime-powers not including 1.

Cf. A001597, A024619, A053289, A053707, A059305, A064113, A366833, A376596, A376597, A376598, A377051, A377054.

Programs

Mathematica

Table[NestWhile[#+1&,Prime[n]+1,!PrimePowerQ[#]&]-Prime[n],{n,100}]

Python

from itertools import count
from sympy import prime, factorint
def A377281(n): return -(p:=prime(n))+next(filter(lambda m:len(factorint(m))<=1, count(p+1))) # Chai Wah Wu, Oct 25 2024

Formula

a(n) = A000015(prime(n)) - prime(n).

a(n) = A345531(n) - prime(n).

a(n) = A377282(prime(n)).