A363597 Union of prime powers and numbers that are not squarefree.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 100, 101, 103

Offset: 1

Author

Michael De Vlieger, Aug 15 2023

Keywords

• nonn
• easy

Comments

Numbers that are prime powers p^m, m >= 0, or products of multiple powers of distinct primes p^m where at least 1 prime power p^m is such that m > 1.

Let N = A000027. Analogous to the following sequences:

A002808 = N \ {{1} U A000040} = {1} U A024619 U A013929,

A303554 = N \ A126706 = A000961 U A005117, and

A085961 = N \ {{1} U A246547} = {A005117 U A024619} \ {1}.

Examples

			1 is in the sequence because it is the empty product.
Prime p is in the sequence because it is not a composite squarefree number.
Numbers k that have prime power factors p^m | k where at least one prime power factor is such that m > 1 are in the sequence because they are not squarefree composites. Examples include 8, 9, 12, 20, and 36.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000961, A013929, A120944, A126706, A246547.

Programs

Mathematica

Select[Range[103], Nand[SquareFreeQ[#], CompositeQ[#]] &]

PARI

isok(k) = (k==1) || isprimepower(k) || !issquarefree(k); \\ Michel Marcus, Aug 24 2023

Python

from math import isqrt
from sympy import mobius, primepi
def A363597(n):
    if n==1: return 1
    def f(x): return n-1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x)
    m, k = n-1, f(n-1)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 02 2024

Formula

Complement of A120944, i.e., A000027 \ A120944.

Union of A000961 and A013929.

Union of {1}, A000040, A126706, and A246547.