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%I #39 Jan 26 2020 09:39:47
%S 0,1,2,3,4,3,4,3,4,4,2,5,2,6,4,3,6,2,8,4,4,5,0,6,0,3,4,2,8,4,4,5,0,6,
%T 0,3,4,2,8,4,4,6,0,8,0,4,2,2,8,4,6,5,2,10,4,7,2,4,6,4,6,2,6,10,6,8,0,
%U 4,2,6,4,3,10,6,10,5,2,6,4,8,4,2,10,4,12
%N For n => 1, the number of distinct summands u and v that can be used in the representation of n as u+v, where u and v are two (possibly equal) Ulam numbers A002858.
%C An odd number in the sequence means that there exists the "pseudo-representation" u + u, where u is an Ulam number. For example, a(22)=5 since 22 = 18 + 4 = 16 + 6 = 11 + 11, and the 5 distinct summands 18,4,16,6,11 are Ulam numbers.
%C Note that both 2 and 3 are values for Ulam numbers since, by the previous comment, a value of 3 means that the Ulam number has the additional "pseudo-representation" u + u (see the Examples).
%C It seems that all nonnegative integers occur as values of this sequence.
%H Rémy Sigrist, <a href="/A285884/b285884.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A285884/a285884.txt">C program for A285884</a>
%H Rémy Sigrist, <a href="/A285884/a285884.png">Density plot of the first 2500000 terms</a>
%e a(23) = 0 since 23 can't be written as the sum of two distinct Ulam numbers. This type of numbers are in A033629.
%e a(94) = 1 since 94 = 47 + 47, where 47 is an Ulam number. This type of numbers are in A287611.
%e a(11) = 2 since 11 has the unique representation 11 = 8 + 3, where 8,3 are Ulam numbers. If such n is also an Ulam number (such as 11), then it is in A002858.
%e a(8) = 3 since it has the representation 8 = 6 + 2 and also the additional "pseudo-representation" 8 = 4 + 4, where 6, 2, and 4 are Ulam numbers. If n has such a "pseudo-representation" and is an Ulam number, then it is in A068799.
%o (C) See Links section.
%Y Cf. A002858, A068799, A033629, A287611.
%K nonn
%O 1,3
%A _Enrique Navarrete_, Apr 27 2017
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