A369854 a(n) = number of nonempty subsets S of {1,2,...,n} such that (number of nonprimes in S) = (number of primes in S).

0, 0, 1, 2, 5, 9, 19, 34, 69, 125, 209, 461, 791, 1715, 3002, 5004, 8007, 19447, 31823, 75581, 125969, 203489, 319769, 817189, 1307503, 2042974, 3124549, 4686824, 6906899, 20030009, 30045014, 84672314, 129024479, 193536719, 286097759, 417225899, 600805295

Offset: 0

Author

Clark Kimberling, Feb 03 2024

Keywords

• nonn

Examples

			a(5) = 9 counts these subsets: {1,2}, {1,3}, {1,5}, {2,4}, {3,4}, {4,5}, {1,2,3,4}, {1,2,4,5}, {1,3,4,5}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000040, A000720, A018252, A037031, A369780, A369781, A369853.

Programs

Maple

a:= n-> binomial(n, numtheory[pi](n))-1:
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 03 2024

Mathematica

Map[Length[Select[Map[Commonest, PrimeQ[Rest[Subsets[Range[#]]]]], # == {False, True} || # == {True, False} &]] &, Range[22]]  (* Peter J. C. Moses, Jan 29 2024 *)

Formula

a(n) = A369781(n) - A369853(n).

a(n) = A037031(n) - 1 = binomial(n,pi(n)) - 1. - Alois P. Heinz, Feb 03 2024

Extensions

a(23)-a(36) from Alois P. Heinz, Feb 03 2024