This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A237628 #18 Nov 07 2024 08:30:30
%S A237628 3,15,15,105,105,1155,1155,1365,15015,15015,15015,255255,255255,
%T A237628 596505,4849845,4849845,4849845,10140585,10140585,179444265,229474245,
%U A237628 229474245,242777535,640049865,5898837945,7357214865,7357214865,7357214865,13350001665,196656364905
%N A237628 a(n) is the smallest product of prime numbers such that all numbers from 6 and 2n can be written as the sum of two prime factors (duplication allowed) of a(n).
%C A237628 The prime factors of a(n) make a subset of prime numbers that satisfies the Goldbach Conjecture for even numbers from 6 to 2n.
%H A237628 Lei Zhou, <a href="/A237628/b237628.txt">Table of n, a(n) for n = 3..79</a>
%e A237628 n=4: 2*4=8. 8=3+5. This is the only possible two-prime decomposition which contains prime numbers 3 and 5, while 6=3+3, 3 is an element of set {3,5}. So a(4)=3*5=15.
%e A237628 n=5: 2*5=10. 6=3+3, 8=3+5, 10=5+5. So two selections of prime numbers in set {3,5} (reuse allowed) can be summed into all three numbers 6, 8, and 10. So a(5)=3*5=15.
%e A237628 ...
%e A237628 n=8: 2n=16. We will be able to find two sets, {3,5,7,11} and {3,5,7,13}, that have such feature:
%e A237628 for set {3,5,7,11}, 6=3+3, 8=3+5, 10=5+5, 12=5+7, 14=7+7, and 16=5+11;
%e A237628 for set {3,5,7,13}, 6=3+3, 8=3+5, 10=5+5, 12=5+7, 14=7+7, and 16=3+13.
%e A237628 3*5*7*11=1155, and 3*5*7*13=1365. 1155<1365, so a(8)=1155. Here we did not count set {3,5,7,11,13} which also has the desired feature since the two shorter sets are its subsets such that the products of the elements in the subsets are obviously smaller than the product of elements in this larger set.
%t A237628 a = {{{3}}}; Table[n2 = 2*n; na = {}; la = Last[a]; lo = Length[la]; Do[ok = 0; Do[p1 = la[[i, j]]; p2 = n2 - p1; If[MemberQ[la[[i]], p2], ok = 1], {j, 1, Length[la[[i]]]}];
%t A237628 If[ok == 1, na = Sort[Append[na, la[[i]]]], Do[p1 = la[[i, j]]; p2 = n2 - p1; If[PrimeQ[p2], ng = Sort[Append[la[[i]], p2]]; big = 0; If[Length[na] > 0, Do[If[Intersection[na[[k]], ng] == na[[k]], big = 1], {k, 1, Length[na]}]]; If[big == 0, na = Sort[Append[na, ng]]]], {j, 1, Length[la[[i]]]}]], {i, 1, lo}]; AppendTo[a, na]; b = {};
%t A237628 lna = Length[na]; Do[prd = Times @@ na[[k]]; AppendTo[b, prd], {k, 1, lna}]; Min[b], {n, 4, 32}](*Program lists the 4th item and beyond*)
%Y A237628 Cf. A000040, A002375, A240708.
%K A237628 nonn,hard
%O A237628 3,1
%A A237628 _Lei Zhou_, May 02 2014