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 A067189 #23 Aug 13 2025 23:44:20
%S A067189 22,24,26,30,40,44,52,56,62,98,128
%N A067189 Numbers that can be expressed as the sum of two primes in exactly three ways.
%C A067189 Corresponds to numbers 2m such that A045917(m)=3. Subsequence of A014091. - _Lekraj Beedassy_, Apr 22 2004
%H A067189 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e A067189 26 is a term as 26 = 23+3 = 19+7 = 13+13 are all the three ways to express 26 as a sum of two primes.
%t A067189 y = Select[Flatten@Table[Prime[i] + Prime[j], {i, 500}, {j, 1, i}], # < Prime[500] &]; Select[Union[y], Count[y, #] == 3 &] (* _Robert Price_, Apr 22 2025 *)
%Y A067189 Cf. A023036.
%Y A067189 Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), A067187 (k=1), A067188 (k=2), this sequence (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).
%K A067189 nonn,fini,full
%O A067189 1,1
%A A067189 _Amarnath Murthy_, Jan 10 2002
%E A067189 Extended by Peter Bertok (peter(AT)bertok.com), who finds (Jan 13 2002) that there are no other terms below 10000 and conjectures there are no further terms in this sequence and A067188, A067190, etc.
%E A067189 _R. K. Guy_ (Jan 14 2002) remarks: "I believe that these conjectures follow from a more general one by Hardy & Littlewood (probably in Some problems of 'partitio numerorum' III, on the expression of a number as a sum of primes, Acta Math. 44(1922) 1-70)."