A352233 Numbers that can be expressed as the sum of two primes in exactly 10 ways.

114, 126, 162, 260, 290, 304, 316, 328, 344, 352, 358, 374, 382, 416, 542, 632

Offset: 1

Author

Wesley Ivan Hurt, Mar 08 2022

Keywords

• nonn
• more

Comments

All terms are even. Conjecture: 632 is the last term. Hardy and Littlewood conjectured a growth rate of the number of decompositions for large even numbers (see Conjecture A in page 32 of Hardy and Littlewood reference), implying this sequence is finite. - Chai Wah Wu, Mar 10 2022

Examples

			114 = 5+109 = 7+107 = 11+103 = 13+101 = 17+97 = 31+83 = 41+73 = 43+71 = 47+67 = 53+61.
		

Links

• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Mathematica, volume 44, pages 1-70 (1923).

Crossrefs

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), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), this sequence (k=10).

Programs

Mathematica

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 10 &] (* _Amiram Eldar, Mar 08 2022 *)