A226135 Let abcd... be the decimal expansion of n. Number of iterations of the map n -> f(n) needed to reach a number < 10, where f(n) = a^b + c^d + ... which ends in an exponent or a base according as the number of digits is even or odd.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 5, 2, 21, 2, 1, 1, 1, 3, 2, 3, 6, 8, 19, 6, 1, 1, 2, 5, 21, 3, 4, 12, 17, 4, 1, 1, 3, 2, 3, 5, 4, 15, 4, 3, 1, 1, 7, 2, 4, 14, 16, 4, 16, 4, 1, 1, 5, 6, 3, 2, 5, 11, 13, 15, 1, 1, 5

Offset

0,25

Comments

Inspired by the sequence A133501 (Number of steps for "powertrain" operation to converge when started at n). It is conjectured that the trajectory for each number converges to a single number < 10.

The conjecture is true, since f(x) < x trivially holds for x > 10^10 and I have verified that for every 10 <= x <= 10^10 there is a k such that f^(k)(x) < x, where f^(k) denotes f applied k times. Both the conventions 0^0 = 1 and 0^0 = 0 have been taken into account. - Giovanni Resta, May 28 2013

Links

Michel Lagneau, Table of n, a(n) for n = 0..10000

Example

a(62) = 7 because:
62 -> 6^2 = 36;
36 -> 3^6 = 729;
729 -> 7^2 + 9^1 = 58;
58 -> 5^8 = 390625;
390625 -> 3^9 + 0^6 + 2^5 = 19715;
19715 -> 1^9 + 7^1 + 5^1 = 13;
13 -> 1^3 = 1;
62 -> 36 -> 729 -> 58 -> 390625 -> 19715 -> 13 -> 1 with 7 iterations.

Maple

A133501:= proc(n)
     local a, i, n1, n2, t1, t2;
     n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=0;
        for i from 0 to floor(t2/2)-1 do
         a := a+t1[t2-2*i]^t1[t2-2*i-1];
       od:
       if t2 mod 2 = 1 then
       a:=a+t1[1]; fi; RETURN(n2*a); end;
A226135:= proc(n)
    local traj , c;
    traj := n ;
    c := [n] ;
    while true do
       traj := A133501(traj) ;
       if member(traj, c) then
       return nops(c)-1 ;
       end if;
       c := [op(c), traj] ;
    end do:
end proc:
seq(A226135(n), n=0..100) ;
# second Maple program:
f:= n-> `if`(n<10, n, `if`(is(length(n), odd), f(10*n+1),
               iquo(irem(n, 100, 'r'), 10, 'h')^h+f(r))):
a:= n-> `if`(n<10, 0, 1+a(f(n))):
seq(a(n), n=0..100);  # Alois P. Heinz, May 27 2013

Crossrefs

Cf. A000312, A031348, A031349, A045503, A133500, A225974.

Sequence in context: A271654 A271622 A324505 * A284464 A038041 A197591

Adjacent sequences: A226132 A226133 A226134 * A226136 A226137 A226138

Keyword

nonn,base

Author

Michel Lagneau, May 27 2013

Status

approved