A369781
a(n) = number of nonempty subsets S of {1,2,...,n} such that (number of primes in S) <= (number of nonprimes in S).
Original entry on oeis.org
0, 1, 2, 3, 10, 15, 41, 63, 162, 381, 847, 1485, 3301, 5811, 12910, 27823, 58650, 109293, 230963, 430103, 910595, 1898711, 3913703, 7507637, 15505589, 31746650, 64574876, 130712028, 263644132, 520381365, 1050777736, 2071510458, 4187373082, 8439258405, 16971077851
Offset: 0
a(4) = 10 enumerates these subsets: {1}, {4}, {1,2}, {1,3}, {1,4}, {2,4}, {3,4}, {1,3,4}, {1,2,4}, {1,2,3,4}.
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Map[Length[Select[Map[Commonest, PrimeQ[Rest[Subsets[Range[#]]]]], # != {True} &]] &, Range[22]] (* Peter J. C. Moses, Jan 29 2024 *)
A369853
a(n) = number of subsets of {1,2,...,n} that contain more nonprimes than primes.
Original entry on oeis.org
0, 1, 1, 1, 5, 6, 22, 29, 93, 256, 638, 1024, 2510, 4096, 9908, 22819, 50643, 89846, 199140, 354522, 784626, 1695222, 3593934, 6690448, 14198086, 29703676, 61450327, 126025204, 256737233, 500351356, 1020732722, 1986838144, 4058348603, 8245721686, 16684980092
Offset: 0
a(4) = 5 enumerates these subsets: {1}, {4}, {1,4}, {1,2,4}, {1,3,4}.
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b:= proc(n, t) option remember; `if`(n=0, `if`(t<0, 1, 0),
b(n-1, t)+b(n-1, t+`if`(isprime(n), 1, -1)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..34); # Alois P. Heinz, Feb 03 2024
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Map[Length[Select[Map[Commonest, PrimeQ[Rest[Subsets[Range[#]]]]], # == {False, True} || # == {True, False} &]] &, Range[22]] (* Peter J. C. Moses, Jan 29 2024 *)
A369854
a(n) = number of nonempty subsets S of {1,2,...,n} such that (number of nonprimes in S) = (number of primes in S).
Original entry on oeis.org
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
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}.
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a:= n-> binomial(n, numtheory[pi](n))-1:
seq(a(n), n=0..36); # Alois P. Heinz, Feb 03 2024
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Map[Length[Select[Map[Commonest, PrimeQ[Rest[Subsets[Range[#]]]]], # == {False, True} || # == {True, False} &]] &, Range[22]] (* Peter J. C. Moses, Jan 29 2024 *)
Showing 1-3 of 3 results.