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

Author

Michel Lagneau, Nov 22 2010

Keywords

• nonn
• base

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}.

Examples

			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.
		

References

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

Crossrefs

Cf. A061602.

Programs

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] ];

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) ;

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