A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

5, 4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1

Offset: 1

Author

Gus Wiseman, Aug 31 2024

Keywords

• nonn

Comments

Non-prime-powers (exclusive) are listed by A361102.

Warning: For this sequence, 1 is not a prime-power but is a non-prime-power.

Examples

			The 6th non-prime-power (exclusive) is 15, and the 7th is 18, so a(6) = 3.
		

Crossrefs

For prime-powers (A000961, A246655) we have A057820, gaps A093555.

For perfect powers (A001597) we have A053289.

For nonprime numbers (A002808) we have A073783.

For squarefree numbers (A005117) we have A076259.

First differences of A361102, inclusive A024619.

Positions of 1's are A375713.

If 1 is considered a prime power we have A375735.

Runs of non-prime-powers:

- length: A110969

- first: A373676

- last: A373677

- sum: A373678

A000040 lists all of the primes, differences A001223.

A007916 lists non-perfect-powers, differences A375706.

A013929 lists the nonsquarefree numbers, differences A078147.

Prime-power runs: A373675, min A373673, max A373674, length A174965.

Prime-power antiruns: A373576, min A120430, max A006549, length A373671.

Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.

Cf. A046933, A061399, A176246, A251092, A375709.

Programs

Mathematica

Differences[Select[Range[100],!PrimePowerQ[#]&]]

Python

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375708(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 09 2024