A369853 - OEIS

%I #13 Feb 03 2024 19:55:28

%S 0,1,1,1,5,6,22,29,93,256,638,1024,2510,4096,9908,22819,50643,89846,

%T 199140,354522,784626,1695222,3593934,6690448,14198086,29703676,

%U 61450327,126025204,256737233,500351356,1020732722,1986838144,4058348603,8245721686,16684980092

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

%H Alois P. Heinz, <a href="/A369853/b369853.txt">Table of n, a(n) for n = 0..2000</a>

%F a(n) = A369781(n) - A369854(n).

%e a(4) = 5 enumerates these subsets: {1}, {4}, {1,4}, {1,2,4}, {1,3,4}.

%p b:= proc(n, t) option remember; `if`(n=0, `if`(t<0, 1, 0),

%p       b(n-1, t)+b(n-1, t+`if`(isprime(n), 1, -1)))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..34);  # _Alois P. Heinz_, Feb 03 2024

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

%Y Cf. A000040, A018252, A369780, A369781, A369854.

%K nonn

%O 0,5

%A _Clark Kimberling_, Feb 03 2024

%E a(23)-a(34) from _Alois P. Heinz_, Feb 03 2024