1,17

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 semiprimes (A001358) or products of two powers of primes (A000961 except 1). Number of ways of representing 2n-1 as a + b + c where omega(a) = omega(b) = omega(c) = 2. Number of ways of representing 2n-1 as a + b + c where A001221(a) = A001221(b) A001221(c) = 2.

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.

Cf. A000961, A001358, A112799, A112800, A112802.

Sequence in context: A007294 A053282 A001584 this_sequence A089873 A096323 A035682

Adjacent sequences: A112798 A112799 A112800 this_sequence A112802 A112803 A112804

nonn

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