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A112802
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Number of ways of representing 2n-1 as sum of three integers with 3 distinct prime factors.
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5
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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, 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, 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
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OFFSET
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1,107
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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isA033992 := proc(n)
numtheory[factorset](n) ;
if nops(%) = 3 then
true;
else
false;
end if;
end proc:
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:
local a, i, j, p, q, r, n2;
n2 := 2*n-1 ;
a := 0 ;
for i from 1 do
if 3*p > n2 then
return a;
else
for j from i do
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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