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 (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.

Sequence in context: A344085 A019565 A309840 * A274608 A319680 A350337

Adjacent sequences: A340313 A340314 A340315 * A340317 A340318 A340319

Keyword

nonn,tabl,changed

Author

Peter Dolland, Jan 04 2021

Status

approved