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

1,1

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.

Links

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

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.

Example

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.

Prog

(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

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.

Sequence in context: A151410 A374568 A230050 * A278459 A278461 A372580

Adjacent sequences: A340464 A340465 A340466 * A340468 A340469 A340470

Keyword

nonn

Author

Alois P. Heinz, Jan 08 2021

Status

approved