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

Author

Peter Dolland, Jan 04 2021

Keywords

• nonn
• tabl

Comments

This is a permutation of all squarefree numbers > 1.

Examples

			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
		

Crossrefs

Cf. A005117 (squarefree numbers), A072047 (number of prime factors), A340313 (indexing), A078840 (all natural numbers, not only squarefree).

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.

Programs

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

Formula

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