A214301 Smallest limiting value of n under iteration of "Sum of its digits raised to its digits power" (A045512).

1, 288, 288, 288, 288, 288, 288, 288, 288, 1, 288, 288, 288, 288, 50119, 3439, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 3439, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 288, 50119

Offset

1,2

Comments

Numbers that yield a fixed value after adding its digits raised to its digits power are called "Munchhausen Numbers" (A046253); i.e. 3435 = 3^3 + 4^4 + 3^3 + 5^5.

Since (9^9)*n < 10^n for n > 12; every initial value should eventually reach a "Munchhausen Number" or a cycle.

This sequence assigns to n that Munchhausen number or the lowest member of the cycle.

About 80% of the numbers reach the 288 cycle; 15% the 3439 cycle; 5% the 50119 cycle and less than 0.1% the fixed value 438579088; only few sporadic numbers reach 1 or 3435.

It has been proved that there are only four Munchhausen numbers (0, 1, 3435 and 438579088) with the convention of 0^0 = 0.

Open questions: are there any other cycles than those described here? What is the % of numbers < 10^n reaching a specific limiting value for n = 1, 2, 3, etc...?

Links

Table of n, a(n) for n=1..51.

Eric Weisstein's World of Mathematics, Munchhausen Number.

Mathematica

f[n_] := Module[{d = IntegerDigits[n]}, Sum[If[i == 0, 0, i^i], {i, d}]]; Table[s = NestWhileList[f, n, UnsameQ[##] &, All]; Min[Drop[s, Position[s, s[[-1]], 1, 1][[1, 1]]]], {n, 100}] (* T. D. Noe, Jul 12 2012 *)

Crossrefs

Cf. A046253, A045512.

Sequence in context: A334008 A035882 A244512 * A296902 A371948 A287116

Adjacent sequences: A214298 A214299 A214300 * A214302 A214303 A214304

Keyword

base,nonn

Author

Sergio Pimentel, Jul 11 2012

Status

approved