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

1,2

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}.

Links

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

Formula

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

Union of A000961 and A013929.

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A374473 A362618 A371782 * A212165 A367685 A344586

Adjacent sequences: A363594 A363595 A363596 * A363598 A363599 A363600

Keyword

nonn,easy

Author

Michael De Vlieger, Aug 15 2023

Status

approved