A166623 -id:A166623 - cp's OEIS frontend

A046253 Equal to the sum of its nonzero digits raised to its own power.

0, 1, 3435, 438579088

Offset: 1

Patrick De Geest, May 15 1998

A variant of Münchausen numbers, cf. A166623.

The sequence is finite, because the sum can't exceed 9^9*L < 10^9*L, where L is the number of digits, and for L > 10 this is less than the number N >= 10^(L-1). - M. F. Hasler, Oct 01 2024

			3435 = 3^3 + 4^4 + 3^3 + 5^5.

J. S. Madachy, "Madachy's Mathematical Recreations", Dover N.Y., pp. 163-175.
C. A. Pickover, "Keys to Infinity", Wiley 1995, Ch. 22, pp. 169-171.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 37.
David Wells, "Curious and Interesting Numbers", Penguin 1988, pp. 169, 190.

Devin Akman, Munchausen Numbers Redux, Missouri J. Math. Sci. 30 (2018), no. 1, 1--4.
Geoff Bailey, C program for the sequence (cf. Hutchens link for more info), Aug. 1998.
Daan van Berkel, On a curious property of 3435, arXiv:0911.3038 [math.HO], 2009.
Jason Hutchens, power summation (originally at ciips.ee.uwa.edu.au/~hutch), 1997.
Eric Weisstein's World of Mathematics, Münchhausen Number.

Cf. A032799, A166623.

Fixed points of A045512. See also A045503 (includes zero digits).

C
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// See Bailey and Hutchens links
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Select[Range[0,10000],Total[#^#&/@DeleteCases[IntegerDigits@#,0]]==#&]  (* Giorgos Kalogeropoulos, May 08 2019 *)

select( {is_A046253(n)=n==A045512(n)}, [0..10^4]) \\ To find the 4th solution, multiply the set by 51817. - M. F. Hasler, Oct 01 2024

A193163 Irregular table read by rows, in which row n lists the factorions in base n, for n >= 2.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 7, 1, 2, 49, 1, 2, 25, 26, 1, 2, 1, 2, 1, 2, 41282, 1, 2, 145, 40585, 1, 2, 26, 48, 40472, 1, 2, 1, 2, 519326767, 1, 2, 12973363226, 1, 2, 1441, 1442, 1, 2, 2615428934649, 1, 2, 40465, 43153254185213, 43153254226251, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 2

Karl W. Heuer, Aug 06 2011

Because 1 and 2 are factorions in any base, these mark the start of a new row.

			49 is in row 5 because 49 = 1! + 4! + 4! and is "144" in base 5.
The first few rows are:
1, 2 (binary)
1, 2 (ternary)
1, 2, 7 (quartal)
1, 2, 49 (quintal)
1, 2, 25, 26 (hexal)
1, 2 (heptal)
1, 2 (octal)
1, 2, 41282 (nonal)
1, 2, 145, 40585 (decimal)
1, 2, 26, 48, 40472 (undecimal)
1, 2 (duodecimal)
1, 2, 519326767, etc.

Karl W. Heuer, Rows n = 2..39, flattened (each row starts with 1 and 2)
Eric Weisstein's World of Mathematics, Factorion

Cf. A014080, A166623.

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);

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);

A279813 Number of Münchhausen numbers in base n.

Original entry on oeis.org

2, 3, 3, 1, 3, 2, 1, 4, 2, 3, 2, 5, 5, 6, 3, 5, 4, 1, 3, 3, 2, 1, 2, 3, 3, 1, 4, 1, 2, 3, 1, 3, 2, 1

Offset: 2

Eric M. Schmidt, Dec 24 2016

See A166623 for a table of Münchhausen numbers in base n.

John D. Cook, Münchausen numbers (2016)
Daan van Berkel, On a curious property of 3435, arXiv:0911.3038 [math.HO], 2009.

Cf. A166623.

cp's OEIS Frontend

A046253 Equal to the sum of its nonzero digits raised to its own power.

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A193163 Irregular table read by rows, in which row n lists the factorions in base n, for n >= 2.

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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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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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Maple

A279813 Number of Münchhausen numbers in base n.

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