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A007963
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Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes.
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12
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0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 6, 8, 7, 9, 10, 10, 10, 11, 12, 12, 14, 16, 14, 16, 16, 16, 18, 20, 20, 20, 21, 21, 21, 27, 24, 25, 28, 27, 28, 33, 29, 32, 35, 34, 30, 37, 36, 34, 42, 38, 36, 46, 42, 42, 50, 46, 47, 53, 50, 45, 56, 54, 46, 62, 53, 48, 64, 59, 55, 68, 61, 59, 68
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OFFSET
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0,7
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COMMENTS
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Ways of writing 2n+1 as p+q+r where p,q,r are odd primes with p <= q <= r.
The two papers of Helfgott appear to provide a proof of the Odd Goldbach Conjecture that every odd number greater than five is the sum of three primes. (The paper is still being reviewed.) - Peter Luschny, May 18 2013; N. J. A. Sloane, May 19 2013
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REFERENCES
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George E. Andrews, Number Theory (NY, Dover, 1994), page 111.
Ivars Peterson, The Mathematical Tourist (NY, W. H. Freeman, 1998, pages 35-37.
Paulo Ribenboim, "VI, Goldbach's famous conjecture," The New Book of Prime Number Records, 3rd ed. (NY, Springer, 1996), pages 291-299.
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LINKS
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EXAMPLE
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a(10) = 4 because 21 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7.
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MAPLE
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local a, i, j, k, p, q, r ;
a := 0 ;
for i from 2 do
p := ithprime(i) ;
for j from i do
q := ithprime(j) ;
for k from j do
r := ithprime(k) ;
if p+q+r = 2*n+1 then
a := a+1 ;
elif p+q+r > 2*n+1 then
break;
end if;
end do:
if p+2*q > 2*n+1 then
break;
end if;
end do:
if 3*p > 2*n+1 then
break;
end if;
end do:
return a;
end proc:
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MATHEMATICA
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nn = 75; ps = Prime[Range[2, nn + 1]]; c = Flatten[Table[If[i >= j >= k, i + j + k, 0], {i, ps}, {j, ps}, {k, ps}]]; Join[{0, 0, 0, 0}, Transpose[Take[Rest[Sort[Tally[c]]], nn+2]][[2]]] (* T. D. Noe, Apr 08 2014 *)
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PROG
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(Sage)
c = 0
for p in Partitions(n, length = 3):
b = True
for t in p:
b = is_prime(t) and t > 2
if not b: break
if b : c = c + 1
return c
(Perl) use ntheory ":all"; sub a007963 { my($n, $c)=(shift, 0); forpart { $c++ if vecall { is_prime($_) } @_; } $n, {n=>3, amin=>3}; $c; }
say "$_ ", a007963(2*$_+1) for 0..100; # Dana Jacobsen, Mar 19 2017
(PARI) a(n)=my(k=2*n+1, s, t); forprime(p=(k+2)\3, k-6, t=k-p; forprime(q=t\2, min(t-3, p), if(isprime(t-q), s++))); s \\ Charles R Greathouse IV, Mar 20 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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R. Muller
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EXTENSIONS
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STATUS
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approved
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