A226036 Let abc... be the decimal expansion of n. a(n) is the number of iterations of the map n -> f(n) needed to reach the last number of the cycle, where f(n) = a^a + b^b + c^c + ...

1, 0, 58, 66, 57, 104, 46, 70, 144, 98, 59, 59, 105, 70, 66, 107, 102, 46, 150, 124, 105, 105, 145, 71, 146, 47, 145, 65, 69, 115, 70, 70, 71, 152, 142, 104, 106, 106, 103, 44, 66, 66, 146, 142, 189, 151, 50, 62, 141, 101, 107, 107, 47, 104, 151, 102, 186, 76

Offset: 0

Author

Michel Lagneau, May 24 2013

Keywords

• nonn
• base

Comments

Additive persistence with powers of decimal digits: number of steps for "add digit(i) ^ digit(i)" operation to stabilize when started at n.

Or number of distinct values obtained by iterating n -> A045503(n).

We take 0^0 = 1.

It is conjectured that the trajectory for every number converges to a single number. The growth of a(n) is very slow; for example, a(457) = 211, a(10337) = 213, a(16669) = 214, ...

Examples

			a(0) = 1 because 0 -> 0^0 = 1 with 1 iteration;
a(1) = 0 because 1 -> 1^1 => 0 iteration;
a(354) = 4 because:
354 -> 3^3 + 5^5 + 4^4 = 3408;
3408 -> 3^3 + 4^4 + 0^0 + 8^8 = 16777500;
16777500 -> 1^1 + 6^6 + 7^7 + 7^7 + 7^7 + 5^5 + 0^0 + 0^0 = 2520413;
2520413 -> 2^2 + 5^5 + 2^2 + 0^0 + 4^4 + 1^1 + 3^3 = 3418 and
3418 is the last number of the cycle because 3418 -> 16777500 is already in the trajectory. We obtain 4 iterations: 354 -> 3408 -> 16777500 -> 2520413 -> 3418.
		

Links

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

Crossrefs

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

Programs

Maple

A000312:=proc(n)
    if n = 0 then 1;
    else add(d^d, d=convert(n, base, 10)) ;
    end if;
end proc:
A226036:= proc(n)
    local traj , c;
    traj := n ;
    c := [n] ;
    while true do
       traj := A000312(traj) ;
       if member(traj, c) then
       return nops(c)-1 ;
       end if;
       c := [op(c), traj] ;
    end do:
end proc:
seq(A226036(n), n=0..100) ;

Mathematica

Unprotect[Power]; 0^0 = 1; Protect[Power]; f[n_] := (cnt++; id = IntegerDigits[n]; Total[id^id]); a[n_] := (cnt = 0; NestWhile[f, n, UnsameQ, All]; cnt-1); Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 24 2013 *)