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Number of partitions of n into two parts with the smaller part prime and the larger part nonprime.
2

%I #14 Apr 28 2020 00:48:53

%S 0,0,0,0,0,1,1,1,1,1,3,2,2,2,3,2,4,2,3,2,3,2,5,2,4,3,6,4,6,3,5,4,5,3,

%T 7,3,7,6,7,5,8,4,7,5,7,5,9,4,8,5,9,6,9,4,8,6,9,6,10,4,9,8,10,6,11,5,

%U 11,9,10,6,11,5,10,7,11,7,12,5,12,8,11,8

%N Number of partitions of n into two parts with the smaller part prime and the larger part nonprime.

%H Harvey P. Dale, <a href="/A302481/b302481.txt">Table of n, a(n) for n = 1..1000</a>

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

%F a(n) = Sum_{i=1..floor(n/2)} c(i) * (1 - c(n-i)), where c is the prime characteristic (A010051).

%e a(11) = 3; 11 = 9+2 = 8+3 = 6+5, smaller parts are prime, larger nonprime.

%t Table[Sum[(1 - (PrimePi[n - i] - PrimePi[n - i - 1])) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[n/2]}], {n, 100}]

%t Table[Count[Boole[PrimeQ[#]&/@IntegerPartitions[n,{2}]],_?(#=={0,1}&)],{n,90}] (* _Harvey P. Dale_, Jan 05 2020 *)

%o (PARI) a(n) = sum(i=1, n\2, isprime(i)*(1-isprime(n-i))); \\ _Michel Marcus_, Apr 09 2018

%Y Cf. A010051, A302480.

%K nonn,easy

%O 1,11

%A _Wesley Ivan Hurt_, Apr 08 2018