A295630 - OEIS

%I #15 May 18 2021 11:44:06

%S 0,0,1,1,1,2,2,2,3,3,5,4,5,5,6,5,8,6,8,7,9,8,11,8,11,10,13,11,14,11,

%T 14,13,15,13,17,13,18,17,18,16,20,16,20,18,21,19,23,18,23,20,25,22,26,

%U 21,26,24,28,25,29,23,29,28,30,26,32,26,33,31,33,29

%N Number of partitions of n into two distinct parts that are not both prime.

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=1..floor((n-1)/2)} 1 - A010051(i)*A010051(n-i).

%e a(12) = 4; The partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). Of these partitions, the parts in (11,1), (10,2), (9,3) and (8,4) are not both prime, so a(12) = 4.

%t Table[Sum[1 - (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}], {n, 80}]

%t Table[Count[IntegerPartitions[n,{2}],_?(#[[1]]!=#[[2]]&&Total[Boole[ PrimeQ[ #]]]<2&)],{n,70}] (* _Harvey P. Dale_, May 18 2021 *)

%Y Cf. A010051, A295629.

%K nonn,easy

%O 1,6

%A _Wesley Ivan Hurt_, Nov 24 2017