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.

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Author

Anders Hellström, Jun 27 2015

Keywords

• nonn

Comments

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

Examples

			After 2, we have to avoid 2^2 = 4, so 3 works.
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.
Note that 16 is the term after 15, because 2^4 and 4^2 are not excluded (since 4 is not in the sequence).
		

Links

• Anders Hellström and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hellström)
• Anders Hellström, Sage program
• Anders Hellström, Mercury program
• Anders Hellström, Ruby program

Crossrefs

Cf. A007916, A259183 (complement).

Equivalent lexicographically earliest sequences for other operations: A000069 (binary exclusive OR), A005408 (addition), A026424 (multiplication).

Programs

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;

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

Formula

a(n) = n + sqrt(n) + O(n^(1/3)). - Charles R Greathouse IV, Aug 18 2015