A361102 - OEIS

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

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OFFSET

1,2

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

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

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 *)

PROG

(SageMath)

def A361102List(upto: int) -> list[int]:

return sorted(Set(1..upto).difference(prime_powers(upto)))

print(A361102List(112)) # Peter Luschny, 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

CROSSREFS

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

Sequence in context: A064040 A168638 A024619 * A335080 A323304 A325411

Adjacent sequences: A361099 A361100 A361101 * A361103 A361104 A361105

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Offset set to 1 by Peter Luschny, May 17 2023

STATUS

approved