A089821 - OEIS

%I #13 Nov 07 2020 05:38:35

%S 0,2,4,8,12,24,32,64,128,256,320,640,768,1536,3072,6144,7168,14336,

%T 16384,32768,65536,131072,147456,294912,589824,1179648,2359296,

%U 4718592,5242880,10485760,11534336,23068672,46137344,92274688,184549376,369098752,402653184

%N Number of subsets of {1,.., n} containing exactly one prime.

%F a(n) = A000720(n)*A089819(n);

%F for n>1: a(n) = A089818(n,1).

%F a(n) = pi(n) * 2^(n-pi(n)), with pi = A000720.

%e a(5)=12 subsets of {1,2,3,4,5} contain exactly one prime: {2}, {3}, {5}, {1,2}, {1,3}, {1,5}, {2,4}, {3,4}, {4,5}, {1,2,4}, {1,3,4} and {1,4,5}.

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

%p      `if`(c<0, 0, b(n-1, c)+b(n-1, c-`if`(isprime(n), 1, 0))))

%p     end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=1..42);  # _Alois P. Heinz_, Dec 19 2019

%t 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]]]];

%t a[n_] := b[n, 1];

%t Array[a, 42] (* _Jean-François Alcover_, Nov 07 2020, after _Alois P. Heinz_ *)

%o (PARI) a(n) = primepi(n) * 2^(n-primepi(n)); \\ _Michel Marcus_, Nov 07 2020

%Y Cf. A000720, A089818, A089819, A089822.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Nov 12 2003