# [seqfan] 4-analogue of Ulam numbers?

Jonathan Post jvospost3 at gmail.com
Sat Jan 1 19:02:29 CET 2011

```Much is known now about
A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least
number > a(n-1) which is a unique sum of two distinct earlier terms.

I've just had a comment approved of a 3-analogue of Ulam numbers:
A007086  Next term is uniquely the sum of 3 earlier terms.
a(1)=1, a(2)=2, a(3)=3, for n>3, a(n) = least number which is a unique
sum of three distinct earlier terms.  Written this way, we see that
this is to 3 as Ulam number A002858 is to 2. - Jonathan Vos Post
(jvospost3(AT)gmail.com)

The next sequence in the supersequence appears, based on quick work by
hand, to be:

4-analogue of Ulam numbers: a(1) = 1; a(2) = 2; a(3) = 3, a(4) = 4,
for n>4, a(n) = least number > a(n-1) which is a unique sum of 4
distinct earlier terms.

1, 2, 3, 4, 10, 16, 17, 18, 19, 22, 63?

n...a(n)
1...1, by definition
2...2, by definition
3...3, by definition
4...4, by definition
5...10, because 1+2+3+4=10 uniquely, and none of {5,6,7,8,9} are in
the set of sums of 4 unique terms of {1,2,3,4} as that set is a
singlet.
6...16, because 1+2+3+10 = 16, and none of {11,12,13,14,15} seem to be
in the set of sums of 4 unique terms of {1,2,3,4,10}
7...17, based on back of the envelope, not confirmed
8...18, ditto
9...19, ditto
10..22, I think
11..63 maybe? 4+8+19+22 = 63; I've found witnesses to nonuniqueness of
23 through 62

This is not hard to code, as Ulam and Knuth did…

```