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%I #10 Jun 09 2014 12:21:21
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,1,1,0,0,2
%N Number of ways of representing 2n-1 as sum of three integers with 3 distinct prime factors.
%C 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.
%H R. J. Mathar, <a href="/A112802/b112802.txt">Table of n, a(n) for n = 1..1290</a>
%H Xianmeng Meng, <a href="http://dx.doi.org/10.1016/j.jnt.2005.04.013">On sums of three integers with a fixed number of prime factors</a>, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
%F 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.
%e 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.
%e 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).
%e 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.
%p isA033992 := proc(n)
%p numtheory[factorset](n) ;
%p if nops(%) = 3 then
%p true;
%p else
%p false;
%p end if;
%p end proc:
%p A033992 := proc(n)
%p option remember;
%p local a;
%p if n = 1 then
%p 30;
%p else
%p for a from procname(n-1)+1 do
%p if isA033992(a) then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc:
%p A112802 := proc(n)
%p local a,i,j,p,q,r,n2;
%p n2 := 2*n-1 ;
%p a := 0 ;
%p for i from 1 do
%p p := A033992(i) ;
%p if 3*p > n2 then
%p return a;
%p else
%p for j from i do
%p q := A033992(j) ;
%p r := n2-p-q ;
%p if r < q then
%p break;
%p end if;
%p if isA033992(r) then
%p a := a+1 ;
%p end if;
%p end do:
%p end if ;
%p end do:
%p end proc:
%p for n from 1 do
%p printf("%d %d\n",n,A112802(n));
%p end do: # _R. J. Mathar_, Jun 09 2014
%Y Cf. A000961, A007304, A112799, A112800, A112801.
%K nonn
%O 1,107
%A _Jonathan Vos Post_ and _Ray Chandler_, Sep 19 2005
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