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

2,2

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.

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.

Eric Weisstein's World of Mathematics, Münchhausen Number.

Example

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;

Prog

(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

Crossrefs

See A046253 for base 10.

Sequence in context: A193603 A059274 A082635 * A348729 A340880 A094510

Adjacent sequences: A166620 A166621 A166622 * A166624 A166625 A166626

Keyword

nonn,base,tabf

Author

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

Extensions

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

More terms from Karl W. Heuer, Aug 06 2011

Status

approved