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 A084422 Number of subsets of integers 1 through n (including the empty set) containing no pair of integers that share a common factor. 29
 1, 2, 4, 8, 12, 24, 28, 56, 72, 104, 116, 232, 248, 496, 544, 616, 728, 1456, 1520, 3040, 3232, 3616, 3872, 7744, 8000, 11168, 11904, 14656, 15488, 30976, 31232, 62464, 69888, 76160, 80256, 89856, 91648, 183296, 192640, 208640, 214272, 428544 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also the number of subsets of {1,...,n} whose product of elements is equal to the least common multiple of elements. - Michel Marcus, Mar 27 2016 REFERENCES Alan Sutcliffe, Divisors and Common Factors in Sets of Integers, awaiting publication. [Apparently unpublished as of 2016] LINKS Alois P. Heinz, Table of n, a(n) for n = 0..220 N. J. Calkin and A. Granville, On the number of coprime-free sets, Number Theory: New York Seminar 1991-1995 (eds. D. Chudnovsky, et.al.), Springer-Verlag (1996). FORMULA a(n) = 1 + Sum_{k=1..A036234(n)} A186974(n,k) if n>0; a(0) = 1. EXAMPLE Exactly 4 of the 2^4=16 subsets of the integers from 1 through 4 contain a pair of integers that share a common factor; these are {2,4}, {1,2,4}, {2,3,4} and {1,2,3,4}. The other 12 subsets do not; hence a(4)=12. MATHEMATICA Prepend[Table[Length@ Select[Rest@ Subsets@ Range@ n, Times @@ # == LCM @@ # &], {n, 22}] + 1, 1] (* Michael De Vlieger, Mar 27 2016 *) PROG (PARI) a(n)=nb = 0; S = vector(n, k, k); for (i = 0, 2^n - 1, ss = vecextract(S, i); if (prod(k=1, #ss, ss[k]) == lcm(ss), nb++); ); nb; \\ Michel Marcus, Mar 27 2016 (PARI) a(n, k=1)=if(n<2, return(n+1)); if(gcd(k, n)==1, a(n-1, n*k)) + a(n-1, k) \\ Charles R Greathouse IV, Aug 24 2016 CROSSREFS Cf. A051026 gives the number of primitive subsets. A087080 gives the number of elements in coprime subsets. A087081 gives the sum of the elements in coprime subsets. Cf. A036234, A186974. Sequence in context: A327480 A330022 A032473 * A175841 A293601 A171647 Adjacent sequences:  A084419 A084420 A084421 * A084423 A084424 A084425 KEYWORD nonn AUTHOR Matthew Vandermast, Jun 26 2003 EXTENSIONS More terms from Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 12 2003 STATUS approved

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Last modified May 23 07:54 EDT 2022. Contains 353961 sequences. (Running on oeis4.)