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%I #21 Mar 11 2022 08:30:30
%S 114,126,162,260,290,304,316,328,344,352,358,374,382,416,542,632
%N Numbers that can be expressed as the sum of two primes in exactly 10 ways.
%C All terms are even. Conjecture: 632 is the last term. Hardy and Littlewood conjectured a grow 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
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Mathematica, volume 44, pages 1-70 (1923).
%e 114 = 5+109 = 7+107 = 11+103 = 13+101 = 17+97 = 31+83 = 41+73 = 43+71 = 47+67 = 53+61.
%t c[n_] := Count[IntegerPartitions[n, {2}], _?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 10 &] (* _Amiram Eldar_, Mar 08 2022 *)
%Y 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).
%K nonn,more
%O 1,1
%A _Wesley Ivan Hurt_, Mar 08 2022
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