OFFSET
1,3
COMMENTS
Lemma: The sequence is finite with all terms in the sequence having at most 22 digits. Proof: Let n be an m-digit natural number in the sequence for some m. Then 10^(m-1)<=n and n<=9+9^2+...9^m = 9(9^m-1)/8<(9^(m+1))/8. Thus 10^(m-1)<(9^(m+1))/8. Taking logarithms of both sides and solving yields m<22.97 QED. Note proof is identical to that for A208130. [Francis J. McDonnell, Apr 14 2012]
Sometimes referred to as disarium numbers. - Dumitru Damian, Jul 22 2024
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 175, p. 55, Ellipses, Paris 2008.
Ken Follett, Code to Zero, Dutton, a Penguin Group, NY 2000, p. 84.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, London, 1986, Entry 175.
LINKS
Rosetta Code, Disarium numbers.
Tim Cross, Problem 2002.1, Student Problems, The Mathematical Gazette, Vol. 86, No. 505 (2002), p. 152.
Eric Weisstein's World of Mathematics, Narcissistic Number.
EXAMPLE
89 = 8^1 + 9^2.
175 = 1^1 + 7^2 + 5^3.
2427 = 2^1 + 4^2 + 2^3 + 7^4.
2646798 = 2^1 + 6^2 + 4^3 + 6^4 + 7^5 + 9^6 + 8^7.
MAPLE
N:= 10: # to get solutions of up to N digits
Branch:= proc(level, sofar)
option remember;
local Res, x, x0, lb, ub, y;
Res:= NULL;
if perm[level] = 1 then x0:= 1 else x0:= 0 fi;
for x from x0 to 9 do
lb:= sofar + b[x, perm[level]] + scmin[level];
ub:= sofar + b[x, perm[level]] + scmax[level];
if lb <= 0 and ub >= 0 then
if level = n then Res:= Res, [x]
else
for y in Branch(level+1, sofar+b[x, perm[level]]) do
Res:= Res, [x, op(y)]
od
fi
fi
od;
[Res]
end:
count:= 0:
for n from 1 to N do
printf("Looking for %d digit solutions\n", n);
forget(Branch);
for j from 1 to n do
for x from 0 to 9 do
b[x, j]:= x^j - x*10^(n-j)
od
od:
for j from 1 to n do
smin[j]:= min(seq(b[x, j], x=0..9));
smax[j]:= max(seq(b[x, j], x=0..9));
od:
perm:= sort([seq(smax[j]-smin[j], j=1..n)], `>`, output=permutation):
for j from 1 to n do
scmin[j]:= add(smin[perm[i]], i=j+1..n);
scmax[j]:= add(smax[perm[i]], i=j+1..n);
end;
for X in Branch(1, 0) do
xx:= add(X[i]*10^(n-perm[i]), i=1..n);
count:= count+1;
A[count]:= xx;
print(xx);
od
od:
seq(A[i], i=1..count); # Robert Israel, Aug 07 2014
MATHEMATICA
f[n_] := Plus @@ (IntegerDigits[n]^Range[ Floor[ Log[10, n] + 1]]); Select[ Range[10^7], f[ # ] == # &] (* Robert G. Wilson v, May 04 2005 *)
Join[{0}, Select[Range[10^7], Total[IntegerDigits[#]^Range[ IntegerLength[ #]]] == #&]] (* Harvey P. Dale, Oct 13 2015 *)
sdcpQ[n_]:=n==Inner[Power, IntegerDigits[n], Range[IntegerLength[n]], Plus]; Join[{0}, Select[Range[27*10^5], sdcpQ]] (* Harvey P. Dale, May 30 2020 *)
PROG
(PARI) for(n=1, 10^22, d=digits(n); s=sum(i=1, #d, d[i]^i); if(s==n, print1(n, ", "))) \\ Derek Orr, Aug 07 2014
CROSSREFS
KEYWORD
nonn,base,fini,full,nice
AUTHOR
Patrick De Geest, May 15 1998
EXTENSIONS
Corrected by Macsy Zhang (macsy(AT)21cn.com), Feb 17 2002
STATUS
approved