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 A112802 Number of ways of representing 2n-1 as sum of three integers with 3 distinct prime factors. 5
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 OFFSET 1,107 COMMENTS Meng proves a remarkable generalization of the Goldbach-Vinogradov classical result that every sufficiently large odd integer N can be partitioned as the sum of three primes N = p1 + p2 + p3. The new proof is that every sufficiently large odd integer N can be partitioned as the sum of three integers N = a + b + c where each of a, b, c has k distinct prime factors for the same k. LINKS R. J. Mathar, Table of n, a(n) for n = 1..1290 Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65. FORMULA Number of ways of representing 2n-1 as sum of three members of A033992. Number of ways of representing 2n-1 as a + b + c where omega(a) = omega(b) = omega(c) = 3, where omega=A001221. EXAMPLE a(83) = 1 because the only partition into three integers each with 3 distinct prime factors of (2*83)-1 = 165 is 165 = 30 + 30 + 105 = (2*3*5) + (2*3*5) + (3*5*7). Coincidentally, 165 itself has three distinct prime factors 165 = 3 * 5 * 11. a(89) = 1 because the only partition into three integers each with 3 distinct prime factors of (2*89)-1 = 177 = 30 + 42 + 105 = (2*3*5) + (2*3*7) + (3*5*7). a(107) = 2 because the two partitions into three integers each with 3 distinct prime factors of (2*107)-1 = 213 are 213 = 30 + 78 + 105 = 42 + 66 + 105. MAPLE isA033992 := proc(n)     numtheory[factorset](n) ;     if nops(%) = 3 then         true;     else         false;     end if; end proc: A033992 := proc(n)     option remember;     local a;     if n = 1 then         30;     else         for a from procname(n-1)+1 do             if isA033992(a) then                 return a;             end if;         end do:     end if; end proc: A112802 := proc(n)     local a, i, j, p, q, r, n2;     n2 := 2*n-1 ;     a := 0 ;     for i from 1 do         p := A033992(i) ;         if 3*p > n2 then             return a;         else             for j from i do                 q := A033992(j) ;                 r := n2-p-q ;                 if r < q then                     break;                 end if;                 if isA033992(r) then                     a := a+1 ;                 end if;             end do:         end if ;     end do: end proc: for n from 1 do     printf("%d %d\n", n, A112802(n)); end do: # R. J. Mathar, Jun 09 2014 CROSSREFS Cf. A000961, A007304, A112799, A112800, A112801. Sequence in context: A084904 A084928 A112316 * A137979 A160338 A216579 Adjacent sequences:  A112799 A112800 A112801 * A112803 A112804 A112805 KEYWORD nonn AUTHOR Jonathan Vos Post and Ray Chandler, Sep 19 2005 STATUS approved

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Last modified March 21 13:51 EDT 2019. Contains 321370 sequences. (Running on oeis4.)