A166623 Irregular triangle read by rows, in which row n lists the Münchhausen numbers in base n, for 2 <= n.

1, 2, 1, 5, 8, 1, 29, 55, 1, 1, 3164, 3416, 1, 3665, 1, 1, 28, 96446, 923362, 1, 3435, 1, 34381388, 34381640, 1, 20017650854, 1, 93367, 30033648031, 8936504649405, 8936504649431, 1, 31, 93344, 17852200903304, 606046687989917

Offset: 2

Author

Daan van Berkel (daan.v.berkel.1980(AT)gmail.com), Oct 18 2009

Keywords

• nonn
• base
• tabf

Comments

Let N = Sum_i d_i b^i be the base b expansion of N. Then N has the Münchhausen property in base b if and only if N = Sum_i (d_i)^(d_i).

Convention: 0^0 = 1.

Examples

			For example: the base 4 representation of 29 is [1,3,1] (29 = 1*4^2 + 3*4^1 + 1*4^0). Furthermore, 29 = 1^1 + 3^3 + 1^1. Therefore 29 has the Münchhausen property in base 4.
Because 1 = 1^1 in every base, a 1 in the sequence signifies a new base.
So the sequence can best be read in the following form:
  1, 2;
  1, 5, 8;
  1, 29, 55;
  1;
  1, 3164, 3416;
  1, 3665;
  1;
  1, 28, 96446, 923362;
  1, 3435;
		

Links

• Karl W. Heuer, Rows n = 2..35, flattened (each row starts with 1)
• John D. Cook, Münchausen numbers (2016)
• Daan van Berkel, On a curious property of 3435, arXiv:0911.3038 [math.HO], 2009.
• Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
• Eric Weisstein's World of Mathematics, Münchhausen Number.

Crossrefs

See A046253 for base 10.

Programs

GAP

next := function(result, n) local i; result[1] := result[1] + 1; i := 1; while result[i] = n do result[i] := 0; i := i + 1; if (i <= Length(result)) then result[i] := result[i] + 1; else Add(result, 1); fi; od; return result; end; munchausen := function(coefficients) local sum, index; sum := 0; for index in coefficients do sum := sum + index^index; od; return sum; end; for m in [2..10] do max := 2*m^m; n := 1; coefficients := [1]; while n <= max do sum := munchausen(coefficients); if (n = sum) then Print(n, "\n"); fi; n := n + 1; coefficients := next(coefficients, m); od; od;

Python

from itertools import combinations_with_replacement
from sympy.ntheory.factor_ import digits
A166623_list = []
for b in range(2,20):
    sublist = []
    for l in range(1,b+2):
        for n in combinations_with_replacement(range(b),l):
            x = sum(d**d for d in n)
            if tuple(sorted(digits(x,b)[1:])) == n:
                sublist.append(x)
    A166623_list.extend(sorted(sublist)) # Chai Wah Wu, May 20 2017

Extensions

Edited (but not checked) by N. J. A. Sloane, Nov 10 2009

More terms from Karl W. Heuer, Aug 06 2011