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A067191
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Numbers that can be expressed as the sum of two primes in exactly five ways.
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12
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48, 54, 64, 70, 74, 76, 82, 86, 94, 104, 124, 136, 148, 158, 164, 188
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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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).
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KEYWORD
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nonn,fini,full
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AUTHOR
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EXTENSIONS
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Corrected and extended by Peter Bertok (peter(AT)bertok.com), Jan 13 2002
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STATUS
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approved
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