A257309 Perfect hyper-4 powers: a^^b, where b <> 1.

0, 1, 4, 16, 27, 256, 3125, 46656, 65536, 823543, 16777216, 387420489, 10000000000, 285311670611, 7625597484987, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979, 104857600000000000000000000

Offset: 1

Author

Natan Arie Consigli, May 07 2015

Keywords

• nonn

Comments

a^^b is the right associative power tower a^a^...^a^a of height b. a^^-1= 0 and a^^0 = 1. We exclude b=1 because otherwise all natural numbers would be in the sequence.

Examples

			Numbers written as power towers include:
5^^2 = 5^5 = 3125;
3^^3 = 3^3^3 = 3^27 = 7625597484987;
2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65536;
0^^5 = 0^0^0^0^0 = 0^0^0^1 = 0^0^0 = 0^1 = 0;
		

Crossrefs

Cf. A004231, A002488.

Programs

Maple

Digits := 200 ;
tpow := proc(a,b,logamax)
    option remember;
    if b = 0 then
        1;
    elif b = 1 then
        a;
    elif b = 2 then
        a^a;
    else
        # log a^procname(a,b-1) = procnmae(a,b-1)*loga
        if evalf(procname(a,b-1,logamax)*log(a)) > evalf(logamax) then
            return -1 ;
        elif procname(a,b-1,logamax)  < 0 then
            return -1 ;
        else
            a^procname(a,b-1,logamax) ;
        end if;
    end if;
end proc:
A257309 := proc(amax)
    local a,n,m,t, logamax;
    a := {0,1} ;
    logamax := evalf(log(amax)) ;
    for n from 2 to amax do
        if n^n > amax then
            break;
        end if;
        for m from 2 do
            t := tpow(n,m,logamax) ;
            if t > amax or t < 0 then
                break;
            elif t <= amax and t > 0 then
                a := a union   {t} ;
            end if;
        end do:
    end do:
    sort(convert(a,list)) ;
end proc:
A257309(10^30) ; # R. J. Mathar, Jun 24 2024