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A166623
Irregular triangle read by rows, in which row n lists the Münchhausen numbers in base n, for 2 <= n.
6
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
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