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A089822
Number of subsets of {1,.., n} containing exactly two primes.
6
0, 0, 2, 4, 12, 24, 48, 96, 192, 384, 640, 1280, 1920, 3840, 7680, 15360, 21504, 43008, 57344, 114688, 229376, 458752, 589824, 1179648, 2359296, 4718592, 9437184, 18874368, 23592960, 47185920, 57671680, 115343360, 230686720
OFFSET
1,3
LINKS
FORMULA
a(n) = (pi(n)*(pi(n)-1)*2^(n-pi(n)))/2, with pi = A000720.
a(n) = A000217(A000720(n)-1)*A089819(n);
for n>2: a(n) = A089818(n,2).
EXAMPLE
a(5)=12 subsets of {1,2,3,4,5} contain exactly two primes: {2,3}, {2,5}, {3,5}, {1,2,3}, {1,2,5}, {1,3,5}, {2,3,4}, {2,4,5}, {3,4,5}, {1,2,3,4}, {1,2,4,5} and {1,3,4,5}.
MAPLE
N:= 100: # for a(1)..a(N)
V:= Vector(N):
p:= 1: n:= 1: pi:= 0:
while n <= N do
p:= nextprime(p);
for n from n to min(N, p-1) do
V[n]:= pi*(pi-1)*2^(n-pi)/2;
od;
pi:= pi+1;
n:= p;
od:
convert(V, list); # Robert Israel, Jul 14 2019
# second Maple program:
b:= proc(n, c) option remember; `if`(n=0, `if`(c=0, 1, 0),
`if`(c<0, 0, b(n-1, c)+b(n-1, c-`if`(isprime(n), 1, 0))))
end:
a:= n-> b(n, 2):
seq(a(n), n=1..42); # Alois P. Heinz, Dec 19 2019
MATHEMATICA
b[n_, c_] := b[n, c] = If[n == 0, If[c == 0, 1, 0], If[c < 0, 0, b[n-1, c] + b[n-1, c - If[PrimeQ[n], 1, 0]]]];
a[n_] := b[n, 2];
Array[a, 42] (* Jean-François Alcover, May 30 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Nov 12 2003
STATUS
approved