A112799 Least odd number such that all greater odd numbers can be represented as sum of three integers with n distinct prime factors (conjectured).

5, 29, 283, 4409, 95539, 2579897, 88149143

Offset: 1

Author

Jonathan Vos Post and Ray Chandler, Sep 19 2005

Keywords

• nonn
• more

Comments

Strangely, the first 5 values of this sequence are all primes. 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.

a(5) = 95539; all odd numbers up to 200000 checked, no larger term found that could not be represented as sum of three integers each with 5 distinct prime factors.

a(1)-a(3): checked odd numbers < 10^5. a(4): checked odd numbers < 10^6. a(5): checked odd numbers < 3*10^6. a(6): checked odd numbers < 3*10^7. a(7): checked odd numbers between 8*10^7 and 2*10^8. [From Donovan Johnson, Feb 04 2009]

Links

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

Crossrefs

Cf. A112800, A112801, A112802.

Extensions

a(6)-a(7) from Donovan Johnson, Feb 04 2009