1,107

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.

Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.

Number of ways of representing 2n-1 as sum of three sphenic numbers (A007304) or products of three powers of primes (A000961 except 1). Number of ways of representing 2n-1 as a + b + c where omega(a) = omega(b) = omega(c) = 3. Number of ways of representing 2n-1 as a + b + c where A001221(a) = A001221(b) A001221(c) = 3.

a(83) = 1 because the only partition into three integers each with 2 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 2 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 2 distinct prime factors of (2*107)-1 = 213 are 213 = 30 + 78 + 105 = 42 + 66 + 105.

Cf. A000961, A007304, A112799, A112800, A112801.

Sequence in context: A071936 A084904 A097516 this_sequence A118269 A137979 A160338

Adjacent sequences: A112799 A112800 A112801 this_sequence A112803 A112804 A112805

nonn

Jonathan Vos Post (jvospost3(AT)gmail.com) and Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 19 2005