A361102 1 together with numbers having at least two distinct prime factors.

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Mar 02 2023

Keywords

• nonn

Comments

This is the union of 1 and A024619. It is the sequence C used in the definition of A360519. Since C is central to the analysis of A360519 it deserves its own entry.

This has the same relationship to A024619 as A000469 does to A120944 for squarefree numbers.

Links

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

Crossrefs

Cf. A000469, A024619, A120944, A126706, A246655, A360519.

Programs

Maple

isa := n -> is(irem(ilcm(seq(1..n-1)), n) = 0):
aList := upto -> select(isa, [seq(1..upto)]):
aList(112); # Peter Luschny, May 17 2023

Mathematica

Select[Range[120], Not@*PrimePowerQ] (* Michael De Vlieger, May 17 2023 *)

Python

from sympy import primepi, integer_nthroot
def A361102(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

SageMath

def A361102List(upto: int) -> list[int]:
    return sorted(Set(1..upto).difference(prime_powers(upto)))
print(A361102List(112))  # Peter Luschny, May 17 2023

Formula

From Peter Luschny and Michael De Vlieger, May 17 2023: (Start)

The sequence is the complement of the prime powers in the positive integers, a = A000027 \ A246655.

k is in this sequence <=> k divides lcm(1, 2, ..., k-1). (End)

This sequence is {1} U { A120944 U A126706 } = {1} U A024619. - Michael De Vlieger, May 17 2023

Extensions

Offset set to 1 by Peter Luschny, May 17 2023