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A112799 Least odd number such that all greater odd numbers can be represented as sum of three integers with n distinct prime factors (conjectured). 3
5, 29, 283, 4409, 95539, 2579897, 88149143 (list; graph; refs; listen; history; text; internal format)



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]


Table of n, a(n) for n=1..7.

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


Cf. A112800, A112801, A112802.

Sequence in context: A292567 A332517 A332469 * A020531 A195228 A226668

Adjacent sequences:  A112796 A112797 A112798 * A112800 A112801 A112802




Jonathan Vos Post and Ray Chandler, Sep 19 2005


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



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