A340467 a(n) is the n-th squarefree number having n prime factors.

2, 10, 66, 462, 4290, 53130, 903210, 17687670, 406816410, 11125544430, 338431883790, 11833068917670, 457077357006270, 20384767656323070, 955041577211912190, 49230430891074322890, 2740956243836856315270, 168909608387276001835590, 11054926927790884163355330

Offset: 1

Author

Alois P. Heinz, Jan 08 2021

Keywords

• nonn

Comments

a(n) is the n-th product of n distinct primes.

All terms are even.

This sequence differs from A073329 which has also nonsquarefree terms.

Examples

			a(1) = A000040(1) = 2.
a(2) = A006881(2) = 10.
a(3) = A007304(3) = 66.
a(4) = A046386(4) = 462.
a(5) = A046387(5) = 4290.
a(6) = A067885(6) = 53130.
a(7) = A123321(7) = 903210.
a(8) = A123322(8) = 17687670.
a(9) = A115343(9) = 406816410.
a(10) = A281222(10) = 11125544430.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..350

Crossrefs

Main diagonal of A340316.

Cf. A001221, A001222, A005117, A070826, A072047, A073329, A101695, A340313.

Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

Programs

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A340467(n):
    if n == 1: return 2
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    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

Formula

a(n) = A340316(n,n).

a(n) = A005117(m) <=> A072047(m) = n = A340313(m).

A001221(a(n)) = A001222(a(n)) = n.

a(n) < A070826(n+1), the least odd number with exactly n distinct prime divisors.