A173447 The number of iterations for the map r -> A061602(r), starting with n, needed to reach the end of the first cycle.

1, 0, 0, 15, 7, 9, 14, 31, 35, 34, 1, 1, 16, 32, 12, 9, 14, 31, 35, 34, 16, 16, 8, 36, 6, 11, 5, 7, 32, 30, 32, 32, 36, 17, 33, 30, 47, 38, 23, 7, 12, 12, 6, 33, 29, 53, 41, 38, 28, 51, 9, 9, 11, 30, 53, 9, 23, 20, 40, 23, 14

Offset

0,4

Comments

The sequence counts the iterations needed to reach a cycle (or loop) of numbers.

The reference (below) proves that the sequence of this numbers is finite and complete.

For n=1, the cycle is (1, 1, 1, ...). For each integer n > 1, the iterations enter one of the following 6 loops:

2, 2, 2, ...

145, 145, 145, ...

169, 363601, 1454, 169, ...

871, 45361, 871, ...

872, 45362, 872, ...

40585, 40585, 40585, ...

The set of numbers occurring in the 7 loops is {1, 2, 145, 169, 871, 872, 1454, 40585, 45361, 45362, 363601}.

References

Jean-Marie De Koninck, Armel Mercier, 1001 problemes en theorie classique des nombres - Entry 258, p. 41. Ellipses, 2004.

Links

Table of n, a(n) for n=0..60.

Example

a(0) = 1 because 0 -> 1;
a(1) = 0 because 1 -> 1;
a(2) = 0 because 2 -> 2;
a(4) = 7 because 4 -> 24 -> 26 -> 722 -> 5044 -> 169 -> 363601 -> 1454, with 7 iterations because 1454 -> 169;
a(40585) = 0 because 40585 -> 40585.

Maple

A061602 := proc(n) if n = 0 then 1; else add(d!, d=convert(n, base, 10)) ; end if; end proc:
A173447 := proc(n) trck := [n] ; while true do m := A061602(op(-1, trck)) ; if m in trck then return nops(trck)-1 ;  else trck := [op(trck), m] ; end if;  end do: end proc:
seq(A173447(n), n=0..50) ;

Prog

(Magma) a061602:=func< n | n eq 0 select 1 else &+[ Factorial(d): d in Intseq(n) ] >;
a173447:=function(n); T:=[]; count:=0; a:=n; while a notin T do Append(~T, a); a:=a061602(a); count+:=1; end while; return count-1; end function; [ a173447(n): n in [0..60] ];

Crossrefs

Cf. A061602.

Sequence in context: A263400 A240909 A133817 * A168211 A131876 A325136

Adjacent sequences: A173444 A173445 A173446 * A173448 A173449 A173450

Keyword

nonn,base

Author

Michel Lagneau, Nov 22 2010

Extensions

Reduced to three lines of numbers; rephrased in terms of A061602; replaced program by a structured version - R. J. Mathar, Nov 23 2010

Terms, offset, and example corrected by D. S. McNeil, Nov 23 2010

Status

approved