cp's OEIS Frontend

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.

%I A340316 #54 Aug 31 2024 18:01:59
%S A340316 2,3,6,5,10,30,7,14,42,210,11,15,66,330,2310,13,21,70,390,2730,30030,
%T A340316 17,22,78,462,3570,39270,510510,19,26,102,510,3990,43890,570570,
%U A340316 9699690,23,33,105,546,4290,46410,690690,11741730,223092870
%N 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.
%C A340316 This is a permutation of all squarefree numbers > 1.
%F A340316 A(A072047(n), A340313(n)) = A005117(n) for n > 1.
%e A340316 First six rows and columns:
%e A340316       2     3     5     7    11    13
%e A340316       6    10    14    15    21    22
%e A340316      30    42    66    70    78   102
%e A340316     210   330   390   462   510   546
%e A340316    2310  2730  3570  3990  4290  4830
%e A340316   30030 39270 43890 46410 51870 53130
%o A340316 (Haskell)
%o A340316 a340316 n k = a340316_row n !! (k-1)
%o A340316 a340316_row n = [a005117_list !! k | k <- [0..], a072047_list !! k == n]
%o A340316 (Python)
%o A340316 from math import prod, isqrt
%o A340316 from sympy import prime, primerange, integer_nthroot, primepi
%o A340316 def A340316_T(n,k):
%o A340316     if n == 1: return prime(k)
%o A340316     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)))
%o A340316     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)))
%o A340316     def bisection(f,kmin=0,kmax=1):
%o A340316         while f(kmax) > kmax: kmax <<= 1
%o A340316         while kmax-kmin > 1:
%o A340316             kmid = kmax+kmin>>1
%o A340316             if f(kmid) <= kmid:
%o A340316                 kmax = kmid
%o A340316             else:
%o A340316                 kmin = kmid
%o A340316         return kmax
%o A340316     return bisection(f) # _Chai Wah Wu_, Aug 31 2024
%Y A340316 Cf. A005117 (squarefree numbers), A072047 (number of prime factors), A340313 (indexing), A078840 (all natural numbers, not only squarefree).
%Y A340316 Rows n=1..10: A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.
%Y A340316 Columns k=1..2: A002110, A306237.
%Y A340316 Main diagonal gives A340467.
%Y A340316 Cf. A358677.
%K A340316 nonn,tabl
%O A340316 1,1
%A A340316 _Peter Dolland_, Jan 04 2021