A067188 Numbers that can be expressed as the (unordered) sum of two primes in exactly two ways.

10, 14, 16, 18, 20, 28, 32, 38, 68

Offset: 1

Author

Amarnath Murthy, Jan 10 2002

Keywords

• nonn
• fini
• full

Comments

Corresponds to numbers 2m such that A045917(m)=2. Subsequence of A014091. - Lekraj Beedassy, Apr 22 2004

Examples

			18 is a term as 18 = 13+5 = 11+7 are the only two ways to express 18 as a sum of two primes.
		

Links

• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A023036.

Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), A067187 (k=1), this sequence (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

Programs

Mathematica

y = Select[Flatten@Table[Prime[i] + Prime[j], {i, 100}, {j, 1, i}], # < Prime[100] &]; Select[Union[y], Count[y, #] == 2 &] (* Robert Price, Apr 22 2025 *)

Extensions

Corrected 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 A067189, A067190, etc.

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)."