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%I #24 Mar 10 2022 10:31:46
%S 48,54,64,70,74,76,82,86,94,104,124,136,148,158,164,188
%N Numbers that can be expressed as the sum of two primes in exactly five ways.
%C 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
%C 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
%C There are no further terms through 50000. - _David Wasserman_, Jan 15 2002
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e 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.
%t upperbound=10^4; range=ConstantArray[0,2*upperbound];
%t primeRange=Prime[Range[PrimePi[upperbound]]];
%t (range[[Plus@@#]]++)&/@(DeleteDuplicates[Sort[#]&/@Tuples[primeRange,2]]);{"upperbound="<>ToString[upperbound],Flatten[Position[Take[range,upperbound],5]]} (* _Hans Rudolf Widmer_, Jul 06 2021 *)
%Y Cf. A002375, A023036.
%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), this sequence (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).
%K nonn,fini,full
%O 1,1
%A _Amarnath Murthy_, Jan 10 2002
%E Corrected and extended by Peter Bertok (peter(AT)bertok.com), Jan 13 2002
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