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A032799
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Numbers n such that n equals the sum of its digits raised to the consecutive powers (1,2,3,...).
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11
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, 12157692622039623539
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OFFSET
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1,3
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COMMENTS
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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]
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,base,fini,full,nice
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AUTHOR
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EXTENSIONS
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Corrected by Macsy Zhang (macsy(AT)21cn.com), Feb 17 2002
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STATUS
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approved
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