A046253 -id:A046253 - cp's OEIS frontend

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

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

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.

			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;

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.

See A046253 for base 10.

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;

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

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

More terms from Karl W. Heuer, Aug 06 2011

A045512 If decimal expansion of n is ab...d, a(n) = a^a + b^b + ... + d^d (ignoring any 0's).

Original entry on oeis.org

0, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 1, 2, 5, 28, 257, 3126, 46657, 823544, 16777217, 387420490, 4, 5, 8, 31, 260, 3129, 46660, 823547, 16777220, 387420493, 27, 28, 31, 54, 283, 3152, 46683, 823570, 16777243, 387420516, 256, 257, 260

Offset: 0

N. J. A. Sloane, Jeff Burch

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A045503.

See A046253 for fixed points.

a:= n-> add(`if`(i=0, 0, i^i), i=convert(n,base,10)):
seq(a(n), n=0..42);  # Alois P. Heinz, Apr 29 2022

p[n_]:=Module[{idn=Select[IntegerDigits[n],#>0&]},Total[idn^idn]]; Array[p, 40,0] (* Harvey P. Dale, Dec 13 2012 *)

apply( {A045512(n)=vecsum([d^d|d<-digits(n),d])}, [0..44]) \\ M. F. Hasler, Oct 01 2024

Checked by Neven Juric (neven.juric(AT)apis-it.hr), Feb 04 2008

A276170 Let n have j digits {d_j, d_(j-1), ..., d_2, d_1}. Sequence lists numbers n such that n = d_j^b_j + d_(j-1)^b_(j-1) + ... + d_2^b_2 + d_1^b_1 for some permutation {b_j, b_(j-1), ..., b_2, b_1} of the digits.

Original entry on oeis.org

1, 1364, 3435, 4155, 4316, 4355, 17463, 48625, 63725, 78215, 117693, 136775, 137456, 137529, 164726, 184746, 196753, 264719, 326617, 326671, 397612, 423858, 516974, 637395, 652812, 653285, 653957, 687523, 834272, 936627, 1374962, 1617349, 1679812, 1683397, 1683514

Offset: 1

Paolo P. Lava, Aug 23 2016

0^0 is not admitted.
652812 is the first number with two essentially different permutations:
6^1 + 5^8 + 2^5 + 8^6 + 1^2 + 2^2 = 6^2 + 5^8 + 2^1 + 8^6 + 1^5 + 2^2 = 652812.

			One of the permutations of {1,3,6,4} is {6,1,4,3} and 1^6+3^1+6^4+4^3 = 1364.

Paolo P. Lava, First 100 terms with applicable permutations

Cf. A046253, A166623.

with(combinat); P:= proc(q) local a,b,c,d,j,k,ok,n;
for n from 1 to q do ok:=1; d:=ilog10(n)+1; a:=convert(n,base,10); b:=permute(a,d);
for k from 1 to nops(b) do c:=0; for j from 1 to d do
if a[j]=0 and b[k][j]=0 then ok:=0; break; else c:=c+a[j]^b[k][j];  fi; od;
if ok=1 then if c=n then print(n); break; fi; fi; od; od; end: P(10^6);

A243507 Consider a decimal number, n, with k digits. n = d(k)10^(k-1) + d(k-1)10^(k-2) + … + d(2)*10 + d_(1). Sequence lists the numbers n that divide s = Sum_{i=1..k} d(i)^d(i).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 63, 64, 93, 377, 643, 699, 760, 2428, 3435, 13073, 46864, 184405, 208858, 1313290, 2326990, 2868720, 2868741, 18273988, 25265859, 33690905, 87889176, 194123725, 589957694

Offset: 1

Paolo P. Lava and Robert G. Wilson v, Jun 05 2014

Since 0^0 is indeterminate, but for all other Xs, X^0 is 1, we define 0^0 here to be 1. (Since 0 does not divide 1, 0 is not a member.)

For Münchhausen numbers (A046253) the ratio is 1. [Paolo P. Lava, Apr 08 2016]

			63 is in the sequence because 6^6+3^3 = 46683 and 46683/63 = 741, an integer.

Cf. A005188, A046253, A243023.

with(numtheory): P:=proc(q) local a,b,k,n; for n from 1 to q do a:=[]; b:=n; while b>0 do a:=[op(a),b mod 10]; b:=trunc(b/10); od; b:=0; for k from 1 to nops(a) do if a[k]=0 then b:=b+1; else b:=b+a[k]^a[k]; fi; od; if type(b/n,integer) then print(n); fi; od; end: P(10^10);

fQ[n_] := Block[{id = IntegerDigits@ n /. {0 -> 1}}, Mod[ Total[ id^id], n] == 0]; k = 1; lst = {}; While[k < 10000000001, If[ fQ@ k, AppendTo[ lst, k]; Print@ k]; k++]; lst

A276241 Let n have j digits {d_j, d_(j-1), ..., d_2, d_1}. Sequence lists numbers n such that R(n) = d_j^b_j + d_(j-1)^b_(j-1) + ... + d_2^b_2 + d_1^b_1 for some permutation {b_j, b_(j-1), ..., b_2, b_1} of the digits, where R(n) is the digits reverse of n.

Original entry on oeis.org

1, 10, 4631, 5343, 5514, 5534, 6134, 36471, 45130, 51287, 52684, 52736, 85200, 176623, 216793, 218256, 272438, 325786, 357691, 396711, 479615, 512870, 577631, 582356, 593736, 627461, 647481, 654731, 716623, 726639, 759356, 858324, 917462, 925731, 945630, 1075785

Offset: 1

Paolo P. Lava, Aug 25 2016

0^0 is not admitted.

			One of the permutations of {4,6,3,1} is {3,4,1,6} and 4^3 + 6^4 + 3^1 + 1^6 = 1364 = R(4631).

Paolo P. Lava, First 100 terms with applicable permutations

Cf. A004086, A046253, A166623, A276170.

with(combinat):  R:=proc(w) local x, y, z; x:=w; y:=0; for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:= proc(q) local a,b,c,d,i,j,k,n,ok,x;
for n from 1 to q do i:=R(n); x:=convert(n,base,10); d:=ilog10(n)+1; b:=permute(x,d); a:={}; ok:=1;
for k from 1 to nops(x) do a:={op(a),x{d-k+1}}; od; for k from 1 to nops(b) do c:=0;
for j from 1 to d do if a{j}=0 and b{k}{j}=0 then ok:=0; break; else c:=c+a{j}^b{k}{j}; fi; od;
if ok=1 then if i=c then print(n); break; fi; fi; od; od;  end: P(10^12);

A334828 Numbers that divide the multiplication of its digits raised to their own powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 25, 36, 64, 96, 125, 128, 135, 162, 175, 216, 250, 256, 375, 378, 384, 432, 486, 567, 576, 625, 648, 672, 675, 729, 735, 756, 768, 784, 864, 875, 972, 1024, 1176, 1250, 1296, 1372, 1715, 1764, 1944, 2048, 2304, 2500, 2744, 2916, 3087, 3125, 3375, 3456, 3645, 3675, 4096

Offset: 1

Scott R. Shannon, May 13 2020

As in A045503 we take 0^0 = 1.

Numbers m that divide A061510(m).

			5 is a term as 5^5 = 3125 which is divisible by 5.
16 is a term as 1^1*6^6 = 46656 which is divisible by 16.
375 is a term as 3^3*7^7*5^5 = 69486440625 which is divisible by 375.
1176 is a term as 1^1*1^1*7^7*6^6 = 38423222208 which is divisible by 1176.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000312, A045503, A046253, A061510, A243507, A108302.

pow[n_] := If[n == 0, 1, n^n]; Select[Range[2^12], Divisible[Times @@ (pow /@ IntegerDigits[#]), #] &] (* Amiram Eldar, May 13 2020 *)

isok(m) = my(d=digits(m)); (prod(k=1, #d, d[k]^d[k]) % m) == 0; \\ Michel Marcus, May 14 2020

A378246 Integers that are equal to the product of their nonzero digits raised to their own power.

Original entry on oeis.org

1, 1024, 12500

Offset: 1

Jason Hammerman, Nov 20 2024

If one accepts 0^0 = 1, the "nonzero" part of the description is unnecessary.
From Michael S. Branicky, Nov 25 2024: (Start)
Terms must be 7-smooth (A002473).
a(4) > 10^200 if it exists. (End)

			1024  = 1^1 * 2^2 * 4^4.
12500 = 1^1 * 2^2 * 5^5.

Jason Hammerman, Python program
Wikipedia, Zero to the power of zero

Fixed points of A061510.
Inspired by A046253.
Cf. A002473.

F[a_]:=If[a==0,1,a^a];Select[Range[10^5],#==Times@@F/@IntegerDigits[#]&] (* James C. McMahon, Dec 14 2024 *)

isok(k) = my(d=digits(k)); k == vecprod(vector(#d, i, d[i]^d[i])); \\ Michel Marcus, Nov 22 2024

Python
```
# See Python program link.
```

from math import prod
def ok(n): return n == prod(d**d for d in map(int, str(n)) if d > 1)
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 24 2024

A048341 Equal to the sum of its digits raised to its reversed digits power (0^0=1).

Original entry on oeis.org

1, 48625, 397612

Offset: 0

Patrick De Geest, Feb 15 1999

			48625 = 4^5 + 8^2 + 6^6 + 2^8 + 5^4.

Harvey Heinz, Narcissistic Numbers
Eric Weisstein's World of Mathematics, Narcissistic Number

Cf. A046253, A032799.

sdrdp[n_]:=Module[{idn=IntegerDigits[n]},Total[#[[1]]^#[[2]]&/@(Thread[ {idn,Reverse[idn]}]/.{0,0}->{1,1})]]; Select[Range[400000],# == sdrdp[#]&] (* Harvey P. Dale, Jul 31 2013 *)

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

Original entry on oeis.org

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

Sergio Pimentel, Jul 11 2012

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...?

Eric Weisstein's World of Mathematics, Munchhausen Number.

Cf. A046253, A045512.

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 *)

cp's OEIS Frontend

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

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A045512 If decimal expansion of n is ab...d, a(n) = a^a + b^b + ... + d^d (ignoring any 0's).

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A276170 Let n have j digits {d_j, d_(j-1), ..., d_2, d_1}. Sequence lists numbers n such that n = d_j^b_j + d_(j-1)^b_(j-1) + ... + d_2^b_2 + d_1^b_1 for some permutation {b_j, b_(j-1), ..., b_2, b_1} of the digits.

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A243507 Consider a decimal number, n, with k digits. n = d(k)*10^(k-1) + d(k-1)*10^(k-2) + … + d(2)*10 + d_(1). Sequence lists the numbers n that divide s = Sum_{i=1..k} d(i)^d(i).

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A276241 Let n have j digits {d_j, d_(j-1), ..., d_2, d_1}. Sequence lists numbers n such that R(n) = d_j^b_j + d_(j-1)^b_(j-1) + ... + d_2^b_2 + d_1^b_1 for some permutation {b_j, b_(j-1), ..., b_2, b_1} of the digits, where R(n) is the digits reverse of n.

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A334828 Numbers that divide the multiplication of its digits raised to their own powers.

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A378246 Integers that are equal to the product of their nonzero digits raised to their own power.

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A048341 Equal to the sum of its digits raised to its reversed digits power (0^0=1).

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A243507 Consider a decimal number, n, with k digits. n = d(k)10^(k-1) + d(k-1)10^(k-2) + … + d(2)*10 + d_(1). Sequence lists the numbers n that divide s = Sum_{i=1..k} d(i)^d(i).