A254225 a(n) = 2[0]2[1]2...2[n-1]2[n]2 where [n] is the n-th hyperoperator.

2, 3, 5, 7, 11, 35

Offset

-1,1

Comments

x[n]y = H_n(x,y) is the aggregation of x and y using the n-th hyperoperator. See A054871 for hyperoperator definitions and key links.

In a(-1) no hyperoperator is applied to 2 so a(-1) = 2. For n>0 calculate the chain a(n) = 2[0]2[1]2...2[n-1]2[n]2. Hyperoperators of higher degree have the higher aggregation priority, so tetration before exponentiation, exponentiation before multiplication, multiplication before addition, etc.

Links

Table of n, a(n) for n=-1..4.

Wikipedia, Hyperoperation.

Index Section Ho-Hy

Formula

a(-1) = 2, a(n) = 2[0]2[1]2...2[n-1]2[n]2.

Example

a(-1)= 2;
a(0) = 2[0]2 = '2 = 3;
a(1) = 2[0]2[1]2 = '2+2 = 5;
a(2) = 2[0]2[1]2[2]2 = '2+2*2 = 7;
a(3) = 2[0]2[1]2[2]2[3]2 = '2+2*2^2 = 11;
a(4) = 2[0]2[1]2[2]2[3]2[4]2 = '2+2*2^2^^2 = '2+2*2^4 = 35;
a(5) = 2[0]2[1]2[2]2[3]2[4]2[5]2 = '2+2*2^2^^2^^^2 = '2+2*2^2^^4 = '2+2*2^2^2^2^2 = '2+2*2^65536 = 3+2^65537 > 4.007*10^19728.

Prog

(PARI) f(x, y, o) = {if (o==4, z=x; for (i=1, y-1, z = x^z); return (z)); if (o==3, return(x^y)); if (o==2, return(x*y)); if (o==1, return(x+y)); }
a(n) = {t = 2; if (n>4, return("too big")); if (n==-1, return(t)); v = vector(n+1, k, t); w = vector(n+1, k, n-k+1); x = v[1]; for (k=1, n+1, if (w[k], x = f(v[k+1], x, w[k]), x = x+1); ); x; } \\ Michel Marcus, Jul 29 2015

Crossrefs

Cf. A000012 (0[0]0[1]...[n]0), A157532 (1[0]1[1]...[n]1), A254310 (3[0]3[1]...[n]3).

Cf. A054871.

Sequence in context: A093487 A067933 A005234 * A334026 A140561 A140553

Adjacent sequences: A254222 A254223 A254224 * A254226 A254227 A254228

Keyword

nonn

Author

Natan Arie Consigli, May 03 2015

Status

approved