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A352296 Smallest number that can be expressed as the sum of two primes in exactly n ways or -1 if no such number exists.

%I A352296 #11 Mar 20 2022 12:15:17
%S A352296 1,4,10,22,34,48,60,78,84,90,114,144,120,168,180,234,246,288,240,210,
%T A352296 324,300,360,474,330,528,576,390,462,480,420,570,510,672,792,756,876,
%U A352296 714,798,690,1038,630,1008,930,780,960,870,924,900,1134,1434,840,990,1302
%N A352296 Smallest number that can be expressed as the sum of two primes in exactly n ways or -1 if no such number exists.
%C A352296 Conjecture: a(n) != -1 for all n.
%C A352296 If n > 0 and a(n) != -1, then a(n) is even.
%C A352296 a(0) = A014092(1)
%C A352296 a(1) = A067187(1)
%C A352296 a(2) = A067188(1)
%C A352296 a(3) = A067189(1)
%C A352296 a(4) = A067190(1)
%C A352296 a(5) = A067191(1)
%C A352296 a(6) = A066722(1)
%C A352296 a(7) = A352229(1)
%C A352296 a(8) = A352230(1)
%C A352296 a(9) = A352231(1)
%C A352296 a(10) = A352233(1)
%t A352296 f[n_] := Count[IntegerPartitions[n, {2}], _?(And @@ PrimeQ[#] &)]; seq[max_] :=  Module[{s = Table[0, {max}], n = 1, c = 0, k}, While[c < max, k = f[n]; If[k < max && s[[k + 1]] == 0, c++; s[[k + 1]] = n]; n++]; s]; seq[50] (* _Amiram Eldar_, Mar 11 2022 *)
%o A352296 (Python)
%o A352296 from itertools import count
%o A352296 from sympy import nextprime
%o A352296 def A352296(n):
%o A352296     if n == 0:
%o A352296         return 1
%o A352296     pset, plist, pmax = {2}, [2], 4
%o A352296     for m in count(2):
%o A352296         if m > pmax:
%o A352296             plist.append(nextprime(plist[-1]))
%o A352296             pset.add(plist[-1])
%o A352296             pmax = plist[-1]+2
%o A352296         c = 0
%o A352296         for p in plist:
%o A352296             if 2*p > m:
%o A352296                 break
%o A352296             if m - p in pset:
%o A352296                 c += 1
%o A352296         if c == n:
%o A352296             return m
%Y A352296 Cf. A014092, A067187, A067188, A067189, A067190, A067191, A066722, A352229, A352230, A352231, A352233.
%Y A352296 Essentially the same as A023036 and A001172.
%K A352296 nonn
%O A352296 0,2
%A A352296 _Chai Wah Wu_, Mar 11 2022