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A259444 a(1)=2. For n>1, a(n) = smallest number > a(n-1) which is different from all the numbers a(i)^a(j) for 1 <= i < n, 1 <= j < n.

%I A259444 #61 Mar 16 2018 05:24:39
%S A259444 2,3,5,6,7,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,26,28,29,30,
%T A259444 31,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,
%U A259444 57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,76
%N A259444 a(1)=2. For n>1, a(n) = smallest number > a(n-1) which is different from all the numbers a(i)^a(j) for 1 <= i < n, 1 <= j < n.
%C A259444 Lexicographically earliest sequence of distinct nonnegative integers with no term being the result of any term raised to the power of any term. - _Peter Munn_, Mar 15 2018
%H A259444 Anders Hellström and Charles R Greathouse IV, <a href="/A259444/b259444.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Hellström)
%H A259444 Anders Hellström, <a href="/A259444/a259444.txt">Sage program</a>
%H A259444 Anders Hellström, <a href="/A259444/a259444_1.txt">Mercury program</a>
%H A259444 Anders Hellström, <a href="/A259444/a259444_2.txt">Ruby program</a>
%F A259444 a(n) = n + sqrt(n) + O(n^(1/3)). - _Charles R Greathouse IV_, Aug 18 2015
%e A259444 After 2, we have to avoid 2^2 = 4, so 3 works.
%e A259444 After 2,3 we have to avoid 2^2=4, 2^3=8, 3^2=9, 3^3=27, so 5 works,also 6, also 7. Then 10, 11 and so on.
%e A259444 Note that 16 is the term after 15, because 2^4 and 4^2 are not excluded (since 4 is not in the sequence).
%o A259444 (PARI) first(m)=my(v=vector(m), x, r, n, s); v[1]=2; for(n=2, m, v[n]=v[n-1]+1; until(x==1, for(r=1, n-1, for(s=1, n-1, if((v[r]^v[s])===v[n]||(v[s]^v[r])===v[n], v[n]++; x=0; break(2), x=1))))); v;
%o A259444 (PARI) list(lim)=if(lim<5, return(if(lim<3,if(lim<2,[],[2]),[2,3]))); my(u=list(max(sqrtint(lim\=1),3)),v=vectorsmall(lim\=1,i,1),r,m,t); for(i=2,#u, v[i]=!!setsearch(u,i)); for(r=1,#u, if(2^u[r]>lim, break); m=0; while(m++<=#u && (t=u[m]^u[r])<=lim, v[t]=0)); u=List(); for(i=2,#v, if(v[i], listput(u,i))); Vec(u) \\ _Charles R Greathouse IV_, Aug 18 2015
%Y A259444 Cf. A007916, A259183 (complement).
%Y A259444 Equivalent lexicographically earliest sequences for other operations: A000069 (binary exclusive OR), A005408 (addition), A026424 (multiplication).
%K A259444 nonn
%O A259444 1,1
%A A259444 _Anders Hellström_, Jun 27 2015