A067191 Numbers that can be expressed as the sum of two primes in exactly five ways.

48, 54, 64, 70, 74, 76, 82, 86, 94, 104, 124, 136, 148, 158, 164, 188

Offset: 1

Author

Amarnath Murthy, Jan 10 2002

Keywords

• nonn
• fini
• full

Comments

There are no other terms below 10000 and I conjecture there are no further terms in this sequence and A067188, A067189, etc. - Peter Bertok (peter(AT)bertok.com), Jan 13 2002

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). - R. K. Guy, Jan 14 2002

There are no further terms through 50000. - David Wasserman, Jan 15 2002

Examples

			70 is a term as 70 = 67 + 3 = 59 + 11 = 53 + 17 = 47 + 23 41 + 29 are all the five ways to express 70 as a sum of two primes.
		

Links

• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A002375, 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), A067190 (k=4), this sequence (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

Programs

Mathematica

upperbound=10^4; range=ConstantArray[0,2*upperbound];
primeRange=Prime[Range[PrimePi[upperbound]]];
(range[[Plus@@#]]++)&/@(DeleteDuplicates[Sort[#]&/@Tuples[primeRange,2]]);{"upperbound="<>ToString[upperbound],Flatten[Position[Take[range,upperbound],5]]} (* Hans Rudolf Widmer, Jul 06 2021 *)

Extensions

Corrected and extended by Peter Bertok (peter(AT)bertok.com), Jan 13 2002