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A302479 Number of partitions of n into two distinct nonprime parts.

%I A302479 #20 Dec 31 2023 22:44:35
%S A302479 0,0,0,0,1,0,1,0,1,2,1,1,2,2,2,3,2,3,3,3,3,5,3,5,4,6,4,6,5,7,6,6,6,9,
%T A302479 6,10,7,8,8,10,8,11,9,10,9,12,9,13,10,13,10,13,11,15,12,14,12,16,13,
%U A302479 18,14,15,14,18,14,20,15,16,16,20,16,21,17,20,17
%N A302479 Number of partitions of n into two distinct nonprime parts.
%H A302479 Antti Karttunen, <a href="/A302479/b302479.txt">Table of n, a(n) for n = 1..10000</a>
%H A302479 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A302479 a(n) = Sum_{i=1..floor((n-1)/2)} (1 - c(i)) * (1 - c(n-i)), where c = A010051.
%F A302479 For n > 0, a(n) = A358638(n) - A005171(n). - _Antti Karttunen_, Nov 25 2022
%e A302479 a(16) = 3; 16 = 15+1 = 12+4 = 10+6, which are distinct nonprimes.
%t A302479 Table[Sum[(1 - PrimePi[n - i] + PrimePi[n - i - 1]) (1 - PrimePi[i] + PrimePi[i - 1]), {i, Floor[(n - 1)/2]}], {n, 100}]
%t A302479 Table[Length[Select[IntegerPartitions[n,{2}],Length[Union[#]]==2&&Boole[PrimeQ[#]]=={0,0}&]],{n,80}] (* _Harvey P. Dale_, Dec 28 2023 *)
%o A302479 (PARI) A302479(n) = sum(k=1,(n-1)\2,!(isprime(k)+isprime(n-k))); \\ _Antti Karttunen_, Nov 25 2022
%Y A302479 Cf. A005171, A010051, A062610, A358638.
%Y A302479 Cf. also A341461, A341462, A341464, A341465, A341466, A341467.
%K A302479 nonn,easy
%O A302479 1,10
%A A302479 _Wesley Ivan Hurt_, Apr 08 2018