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Ulam numbers (Posted on 2017-06-03) Difficulty: 3 of 5
We define recursively the Ulam numbers by setting u1 = 1, u2 = 2, and for each subsequent integer n, we set n equal to the next Ulam number if it can be written uniquely as the sum of two different Ulam numbers; e.g.: u3 = 3, u4 = 4, u5 = 6, etc.

Prove that there are infinitely many Ulam numbers.

Now a D4 BONUS.
3 (=1+2). Find another Ulam number is that is the sum of two consecutive Ulam numbers.

How about a 3rd one?

See The Solution Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
What the OEIS says Comment 5 of 5 |
The Ulam numbers are OEIS A002858.  In the comments: "3 (=1+2) and 131 (=62+69) are the only two Ulam numbers in the first 28 billion Ulam numbers that are the sum of two consecutive Ulam numbers."

Given that the 28 billionth Ulam number is 378,485,625,853 it is possible that there is no third Ulam number that is the sum of two consecutive Ulam numbers.

  Posted by Brian Smith on 2017-06-04 09:42:56
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