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%I #5 May 21 2024 05:27:52
%S 1,2,3,4,16,52,204,255,4259,4262,4265,4855
%N Numbers k that divide the k-th Ulam number.
%C Numbers k such that k | A002858(k).
%C a(13) >= 10^8, if it exists.
%C Based on empirical data its seems that the Ulam numbers have a positive asymptotic density and that A002858(k) ~ 13.5... * k (see A307331 and A346216). If this is true, then this sequence is finite, and it is likely that there are no more terms.
%H <a href="/index/U#Ulam_num">Index entries for Ulam numbers</a>.
%e 16 is a term since A002858(16) = 48 = 3 * 16 is divisible by 16.
%t ulams = {1, 2}; Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[ Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {5000}];
%t Position[ulams/Range[Length[ulams]], _?IntegerQ] // Flatten (* after _Jean-François Alcover_ at A002858 *)
%Y Cf. A002858, A307331, A346216.
%Y Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).
%K nonn,more
%O 1,2
%A _Amiram Eldar_, May 21 2024