A067190 Numbers that can be expressed as the sum of two primes in exactly four ways.

34, 36, 42, 46, 50, 58, 80, 88, 92, 122, 152

Offset: 1

Author

Amarnath Murthy, Jan 10 2002

Keywords

• nonn
• fini
• full

Examples

			36 is a term as 36 = 31 + 5 = 29 + 7 = 23 + 13 = 19 + 17 are all the four ways to express 36 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), A067188 (k=2), A067189 (k=3), this sequence (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, 500}, {j, 1, i}], # <  Prime[500] &]; Select[Union[y], Count[y, #] == 4 &] (* Robert Price, Apr 22 2025 *)

Extensions

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, A067189, etc.

R. K. Guy (Jan 14 2002) remarks: "I believe that these conjectures follow from a more general one by Hardy and 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)."