A084422 Number of subsets of integers 1 through n (including the empty set) containing no pair of integers that share a common factor.

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

Offset: 0

Author

Matthew Vandermast, Jun 26 2003

Keywords

• nonn

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

Examples

			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.
		

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 [Apparently unpublished as of 2016]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).
• Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.

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.

Programs

Mathematica

Prepend[Table[Length@ Select[Rest@ Subsets@ Range@ n, Times @@ # == LCM @@ # &], {n, 22}] + 1, 1] (* Michael De Vlieger, Mar 27 2016 *)

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

Formula

a(n) = 1 + Sum_{k=1..A036234(n)} A186974(n,k) if n>0; a(0) = 1.

Extensions

More terms from Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 12 2003