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

Author

Sergio Pimentel, Jul 11 2012

Keywords

• base
• nonn

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

• Eric Weisstein's World of Mathematics, Munchhausen Number.

Crossrefs

Cf. A046253, A045512.

Programs

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 *)