

A112801


Number of ways of representing 2n1 as sum of three integers, each with two distinct prime factors.


7



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
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OFFSET

1,17


COMMENTS

Meng proves a remarkable generalization of the GoldbachVinogradov 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. 3765.


FORMULA

Number of ways of representing 2n1 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*21; sum(a=6, n\3, if(omega(a)==2, sum(b=a, (na)\2, omega(b)==2 && omega(nab)==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



