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A112801 Number of ways of representing 2n-1 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 (list; graph; refs; listen; history; text; internal format)
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 * A173862 A089873 A275433

Adjacent sequences:  A112798 A112799 A112800 * A112802 A112803 A112804

KEYWORD

nonn,look

AUTHOR

Jonathan Vos Post and Ray Chandler, Sep 19 2005

STATUS

approved

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Last modified March 25 01:30 EDT 2017. Contains 284036 sequences.