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A183168 Number of partitions of n^2 into three distinct primes.

%I A183168 #18 Jun 11 2025 19:45:53
%S A183168 0,0,0,1,2,3,8,2,17,3,41,7,61,6,69,8,152,11,216,6,204,10,383,16,464,
%T A183168 14,447,14,791,18,978,18,822,18,1302,25,1684,27,1387,24,2347,28,2726,
%U A183168 27,2012,34,3620,31,4005,32,3218,44,5274,44,5461,36,4538,43,7429,58
%N A183168 Number of partitions of n^2 into three distinct primes.
%C A183168 For n>3, odd n have (many) more partitions than even n.
%H A183168 Zak Seidov, <a href="/A183168/b183168.txt">Table of n, a(n) for n = 1..200</a>
%e A183168 a(4)=1 because 16=2+3+11,
%e A183168 a(6)=3 because 36=2+3+31=2+5+29=2+11+13.
%t A183168 Table[Count[Union/@IntegerPartitions[n^2,{3}],_?(Length[#]==3&&AllTrue[ #, PrimeQ]&)],{n,60}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, May 02 2018 *)
%o A183168 (PARI) a(n)=my(s);n*=n;forprime(p=n\3,n-4,forprime(q=(n-p)\2+1,min(n-p,p-1),if(isprime(n-p-q),s++)));s \\ _Charles R Greathouse IV_, Aug 27 2012
%o A183168 (Haskell)
%o A183168 a183168 n = z (drop (fromInteger (mod n 2)) a000040_list) (n ^ 2) 3 where
%o A183168    z _      m 1 = if m <= 0 then 0 else a010051 m
%o A183168    z (p:ps) m c = if m <= 2*p then 0 else z ps (m - p) (c - 1) + z ps m c
%o A183168 -- _Reinhard Zumkeller_, Aug 28 2012
%Y A183168 Subsequence of A125688: a(n) = A125688(n^2).
%Y A183168 Cf. A215933, A215934.
%Y A183168 Cf. A000040, A010051.
%Y A183168 Cf. A000607.
%K A183168 nonn
%O A183168 1,5
%A A183168 _Zak Seidov_, Aug 27 2012