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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

%I

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

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

%C 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.

%C 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.

%C 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]

%H Xianmeng Meng, <a href="https://doi.org/10.1016/j.jnt.2005.04.013">On sums of three integers with a fixed number of prime factors</a>, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.

%Y Cf. A112800, A112801, A112802.

%K nonn,more

%O 1,1

%A _Jonathan Vos Post_ and _Ray Chandler_, Sep 19 2005

%E a(6)-a(7) from _Donovan Johnson_, Feb 04 2009

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Last modified June 15 08:14 EDT 2021. Contains 345048 sequences. (Running on oeis4.)