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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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, ...

LINKS

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

EXAMPLE

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.

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-Fran├žois Alcover, May 24 2013 *)

CROSSREFS

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

Sequence in context: A129546 A112826 A224359 * A127024 A010338 A184074

Adjacent sequences:  A226033 A226034 A226035 * A226037 A226038 A226039

KEYWORD

nonn,base

AUTHOR

Michel Lagneau, May 24 2013

STATUS

approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)