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A112802 Number of ways of representing 2n-1 as sum of three integers with 3 distinct prime factors. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A229878 A235145 A266342 A285936 A037281 A143241 A258825

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 November 23 20:23 EST 2017. Contains 295141 sequences.