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%I #14 Nov 28 2020 20:02:53
%S 0,0,1,0,1,2,1,2,3,2,1,2,1,2,3,2,1,2,1,2,3,2,1,2,3,2,3,2,1,2,1,2,3,2,
%T 3,2,1,2,3,2,1,2,1,2,3,2,1,2,3,2,3,2,1,2,3,2,3,2,1,2,1,2,3,2,3,2,1,2,
%U 3,2,1,2,1,2,3,2,3,2,1,2,3,2,1,2,3,2,3,2,1,2,3,2,3,2,3,2,1,2,3,2,1,2,1,2,3,2,1,2,1,2
%N Least number of odd primes that add up to n, or 0 if no such representation is possible.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e a(9) = 3 because 9 = 3 + 3 + 3 is a partition of 9 into 3 odd prime parts and there is no such partition with fewer terms.
%t Block[{f, a}, f[m_] := Block[{s = {Prime@ PrimePi@ m}}, KeySort@ Merge[#, Identity] &@ Reap[Do[If[# <= m, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 3, s = DeleteCases[s, 3]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]][[-1, -1]] ]; a = f[105]; Array[If[KeyExistsQ[a, #], Min@ Map[Length, Lookup[a, #]], 0] &, Max@ Keys@ a]] (* _Michael De Vlieger_, Nov 04 2020 *)
%Y Cf. A051034, A065091.
%K nonn
%O 1,6
%A _Ilya Gutkovskiy_, Nov 04 2020