A340316 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where row n is the increasing list of all squarefree numbers with n primes.

2, 3, 6, 5, 10, 30, 7, 14, 42, 210, 11, 15, 66, 330, 2310, 13, 21, 70, 390, 2730, 30030, 17, 22, 78, 462, 3570, 39270, 510510, 19, 26, 102, 510, 3990, 43890, 570570, 9699690, 23, 33, 105, 546, 4290, 46410, 690690, 11741730, 223092870

Offset

1,1

Comments

This is a permutation of all squarefree numbers > 1.

Links

Table of n, a(n) for n=1..45.

Formula

A(A072047(n), A340313(n)) = A005117(n) for n > 1.

Example

First six rows and columns:
      2     3     5     7    11    13
      6    10    14    15    21    22
     30    42    66    70    78   102
    210   330   390   462   510   546
   2310  2730  3570  3990  4290  4830
  30030 39270 43890 46410 51870 53130

Prog

(Haskell)
a340316 n k = a340316_row n !! (k-1)
a340316_row n = [a005117_list !! k | k <- [0..], a072047_list !! k == n]
(Python)
from math import prod, isqrt
from sympy import prime, primerange, integer_nthroot, primepi
def A340316_T(n, k):
    if n == 1: return prime(k)
    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(k+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

Cf. A005117 (squarefree numbers), A072047 (number of prime factors), A340313 (indexing).

A078840 and A091538 are equivalent tables for all natural numbers, not only squarefree (and with respect to the difference, A392409 looks at relative densities).

Rows n=1..10: A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

Columns k=1..2: A002110, A306237.

Main diagonal gives A340467.

Cf. A358677.

Sequence in context: A019565 A309840 A377566 * A274608 A319680 A350337

Adjacent sequences: A340313 A340314 A340315 * A340317 A340318 A340319

Keyword

nonn,tabl

Author

Peter Dolland, Jan 04 2021

Status

approved