

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) 170).  R. K. Guy, Jan 14 2002
There are no further terms through 50000.  David Wasserman, Jan 15 2002


LINKS

Table of n, a(n) for n=1..16.
Index entries for sequences related to Goldbach conjecture


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



