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

%I A369854 #15 Feb 03 2024 19:59:35
%S A369854 0,0,1,2,5,9,19,34,69,125,209,461,791,1715,3002,5004,8007,19447,31823,
%T A369854 75581,125969,203489,319769,817189,1307503,2042974,3124549,4686824,
%U A369854 6906899,20030009,30045014,84672314,129024479,193536719,286097759,417225899,600805295
%N A369854 a(n) = number of nonempty subsets S of {1,2,...,n} such that (number of nonprimes in S) = (number of primes in S).
%H A369854 Alois P. Heinz, <a href="/A369854/b369854.txt">Table of n, a(n) for n = 0..2000</a>
%F A369854 a(n) = A369781(n) - A369853(n).
%F A369854 a(n) = A037031(n) - 1 = binomial(n,pi(n)) - 1. - _Alois P. Heinz_, Feb 03 2024
%e A369854 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}.
%p A369854 a:= n-> binomial(n, numtheory[pi](n))-1:
%p A369854 seq(a(n), n=0..36);  # _Alois P. Heinz_, Feb 03 2024
%t A369854 Map[Length[Select[Map[Commonest, PrimeQ[Rest[Subsets[Range[#]]]]], # == {False, True} || # == {True, False} &]] &, Range[22]]  (* _Peter J. C. Moses_, Jan 29 2024 *)
%Y A369854 Cf. A000040, A000720, A018252, A037031, A369780, A369781, A369853.
%K A369854 nonn
%O A369854 0,4
%A A369854 _Clark Kimberling_, Feb 03 2024
%E A369854 a(23)-a(36) from _Alois P. Heinz_, Feb 03 2024