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 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. 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified September 12 07:57 EDT 2024. Contains 375850 sequences. (Running on oeis4.)