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 A067191 Numbers that can be expressed as the sum of two primes in exactly five ways. 12
 48, 54, 64, 70, 74, 76, 82, 86, 94, 104, 124, 136, 148, 158, 164, 188 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 LINKS EXAMPLE 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. 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 *) 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). Sequence in context: A357429 A258694 A328738 * A080854 A255267 A345503 Adjacent sequences: A067188 A067189 A067190 * A067192 A067193 A067194 KEYWORD nonn,fini,full AUTHOR Amarnath Murthy, Jan 10 2002 EXTENSIONS Corrected and extended by Peter Bertok (peter(AT)bertok.com), Jan 13 2002 STATUS approved

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Last modified December 5 19:04 EST 2022. Contains 358588 sequences. (Running on oeis4.)