A112801 Number of ways of representing 2n-1 as sum of three integers, each with two distinct prime factors.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 2, 4, 4, 4, 8, 7, 8, 11, 11, 13, 15, 16, 18, 23, 23, 26, 30, 31, 33, 40, 40, 45, 51, 53, 56, 62, 66, 66, 76, 79, 82, 88, 94, 96, 105, 111, 111, 124, 127, 132, 141, 145, 148, 164, 166, 170, 180, 187, 187, 206, 204, 208

Offset

1,17

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.

See A243751 for the range of this sequence, and A243750 for the indices of record values. - M. F. Hasler, Jun 09 2014

Links

R. J. Mathar, Table of n, a(n) for n = 1..1020

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 a + b + c where a<=b<=c are elements of A007774.

Example

a(14) = 1 because the only partition into three integers each with 2 distinct prime factors of (2*14)-1 = 27 is 27 = 6 + 6 + 15 = (2*3) + (2*3) + (3*5).
a(16) = 1 because the only partition into three integers each with 2 distinct prime factors of (2*16)-1 = 31 is 31 = 6 + 10 + 15 = (2*3) + (2*5) + (3*5).
a(17) = 2 because the two partitions into three integers each with 2 distinct prime factors of (2*17)-1 = 33 are 33 = 6 + 6 + 21 = 6 + 12 + 15.

Prog

(PARI) A112801(n)={n=n*2-1; sum(a=6, n\3, if(omega(a)==2, sum(b=a, (n-a)\2, omega(b)==2 && omega(n-a-b)==2)))} \\ M. F. Hasler, Jun 09 2014

Crossrefs

Cf. A000961, A112799, A112800, A112802.

Sequence in context: A001584 A180019 A274496 * A344853 A173862 A089873

Adjacent sequences: A112798 A112799 A112800 * A112802 A112803 A112804

Keyword

nonn,look

Author

Jonathan Vos Post and Ray Chandler, Sep 19 2005

Status

approved