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 A007963 Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes. 12
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS 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 REFERENCES 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. LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012. H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013. H. A. Helfgott, The ternary Goldbach conjecture is true, arxiv:1312.7748 [math.NT], 2013. H. A. Helfgott, The ternary Goldbach problem, arXiv:1404.2224 [math.NT], 2014. F. Smarandache, Only Problems, Not Solutions!. EXAMPLE a(10) = 4 because 21 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7. MAPLE A007963 := proc(n)     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: seq(A007963(n), n=0..30) ; # R. J. Mathar, Sep 06 2014 MATHEMATICA 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 *) PROG (Sage) def A007963(n):     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 [A007963(2*n+1) for n in (0..77)]   # Peter Luschny, May 18 2013 (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 CROSSREFS Cf. A068307, A087916, A294294 (lower bound of scatterplot), A294357, A294358 (records). Sequence in context: A092982 A248868 A030566 * A137222 A077641 A194210 Adjacent sequences:  A007960 A007961 A007962 * A007964 A007965 A007966 KEYWORD nonn AUTHOR R. Muller EXTENSIONS Corrected and extended by David W. Wilson STATUS approved

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Last modified March 23 09:19 EDT 2018. Contains 301104 sequences. (Running on oeis4.)