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A112800 Number of ways of representing 2n-1 as sum of three integers with 1 distinct prime factor. 4
0, 0, 0, 1, 3, 4, 6, 8, 9, 10, 12, 14, 14, 16, 18, 18, 20, 23, 25, 26, 28, 30, 30, 32, 32, 34, 37, 36, 40, 43, 42, 44, 46, 46, 46, 50, 51, 53, 59, 57, 57, 61, 62, 62, 66, 68, 69, 71, 72, 71, 73, 76, 74, 81, 81, 78, 87, 90, 87, 91, 93, 90, 94, 97, 94, 100, 107, 103, 114, 115 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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..1655

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 primes (A000040) or powers of primes (A000961 except 1). Number of ways of representing 2n-1 as a + b + c where omega(a) = omega(b) = omega(c) = 1.

EXAMPLE

a(4) = 1 because the only partition into nontrivial prime powers of (2*4)-1 = 7 is 7 = 2 + 2 + 3.

a(5) = 3 because the 3 partitions into nontrivial prime powers of (2*5)-1 = 9 are 9 = 2 + 2 + 5 = 2 + 3 + 4 = 3 + 3 + 3. The middle one of those partitions has "4" which is not a prime, but is a power of a prime.

a(6) = 4 because the 4 partitions into nontrivial prime powers of (2*6)-1 = 11 are 11 = 2 + 2 + 7 = 2 + 4 + 5 = 3 + 3 + 5.

a(7) = 6 because the 6 partitions into nontrivial prime powers of (2*7)-1 = 13 are 13 = 2 + 2 + 9 = 2 + 3 + 8 = 2 + 4 + 7 = 3 + 3 + 7 = 3 + 5 + 5 = 4 + 4 + 5.

MAPLE

isA000961 := proc(n)

    if n = 1 then

        return true;

    end if;

    numtheory[factorset](n) ;

    if nops(%) = 1 then

        true;

    else

        false;

    end if;

end proc:

A000961 := proc(n)

    option remember;

    local a;

    if n = 1 then

        1;

    else

        for a from procname(n-1)+1 do

            if isA000961(a) then

                return a;

            end if;

        end do:

    end if;

end proc:

A112800 := proc(n)

    local a, i, j, p, q, r, n2;

    n2 := 2*n-1 ;

    a := 0 ;

    for i from 2 do

        p := A000961(i) ;

        if 3*p > n2 then

            return a;

        else

            for j from i do

                q := A000961(j) ;

                r := n2-p-q ;

                if r < q then

                    break;

                end if;

                if isA000961(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, A112800(n));

end do: # R. J. Mathar, Jun 09 2014

CROSSREFS

Cf. A000040, A112799, A112801, A112802.

Sequence in context: A134745 A183867 A182829 * A294456 A062969 A175035

Adjacent sequences:  A112797 A112798 A112799 * A112801 A112802 A112803

KEYWORD

nonn

AUTHOR

Jonathan Vos Post and Ray Chandler, Sep 19 2005

STATUS

approved

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Last modified May 18 23:08 EDT 2021. Contains 344008 sequences. (Running on oeis4.)