A377289 Difference between prime(n) and the previous prime-power (exclusive).

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

Offset: 1

Author

Gus Wiseman, Oct 23 2024

Keywords

• nonn

Examples

			The twelfth prime is 37, with previous prime-power 32, so a(12) = 5.
		

Crossrefs

For powers of two see A013597, A014210, A014234, A244508, A304521.

For prime instead of prime-power we have A075526.

This is the restriction of A276781 (shifted right) to the primes.

For next instead of previous prime-power we have A377281, restriction of A377282.

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.

A065514 gives the greatest prime-power < prime(n).

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

A246655 lists the prime-powers not including 1.

Cf. A002808, A024619, A053707, A059305, A064113, A065890, A366833, A376596, A376597, A377051, A377054.

Programs

Mathematica

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

Python

from sympy import prime, factorint
def A377289(n): return (p:=prime(n))-next(filter(lambda m:len(factorint(m))<=1, range(p-1,0,-1))) # Chai Wah Wu, Oct 25 2024

Formula

a(n) = prime(n) - A031218(prime(n)-1).

a(n) = prime(n) - A065514(n).

a(n) = A276781(prime(n)-1).