A377703 First differences of the sequence A345531(k) = least prime-power greater than the k-th prime.

1, 3, 1, 5, 3, 3, 4, 2, 6, 1, 9, 2, 4, 2, 10, 2, 3, 7, 2, 6, 2, 8, 8, 4, 2, 4, 2, 4, 8, 7, 9, 2, 10, 2, 6, 6, 4, 2, 10, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 13, 7, 6, 2, 6, 4, 2, 6, 18, 4, 2, 4, 14, 6, 6, 6, 4, 6, 2, 12, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6

Offset: 1

Author

Gus Wiseman, Nov 07 2024

Keywords

• nonn

Comments

What is the union of this sequence? In particular, does it contain 17?

Crossrefs

First differences of A345531.

A000961 lists the powers of primes, differences A057820.

A001597 lists the perfect-powers, differences A053289, seconds A376559.

A007916 lists the non-perfect-powers, differences A375706, seconds A376562.

A024619 lists the non-prime-powers, differences A375735, seconds A376599.

A080101 counts prime-powers between primes (exclusive).

A246655 lists the prime-powers, differences A057820 without first term.

A361102 lists the non-powers of primes, differences A375708.

A366833 counts prime-powers between primes, see A053607, A304521, A377057 (positive), A377286 (zero), A377287 (one), A377288 (two).

A377432 counts perfect-powers between primes, see A377434 (one), A377436 (zero), A377466 (multiple).

Cf. A000040, A008864, A031218, A377281, A377282, A377286, A377289, A377468.

Programs

Mathematica

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

Python

from sympy import factorint, prime, nextprime
def A377703(n): return -next(filter(lambda m:len(factorint(m))<=1, count((p:=prime(n))+1)))+next(filter(lambda m:len(factorint(m))<=1, count(nextprime(p)+1))) # Chai Wah Wu, Nov 14 2024