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A032799 Numbers n such that n equals the sum of its digits raised to the consecutive powers (1,2,3,...). 8
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, 12157692622039623539 (list; graph; refs; listen; history; text; internal format)
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]

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

Table of n, a(n) for n=1..20.

Eric Weisstein's World of Mathematics, Narcissistic Number

FORMULA

Let a(n)=Sum_{x=0,floor(log(n))} (floor(n/10^x)-10*floor(n/10^(x + 1)))^(floor(log10(n) + 1) - x) - n. If a(n)=0, then n is a number of this sequence. - José de Jesús Camacho Medina, Aug 07 2014

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 *)

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

Sequence in context: A135480 A228326 A098766 * A208130 A160343 A239085

Adjacent sequences:  A032796 A032797 A032798 * A032800 A032801 A032802

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

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Last modified September 17 23:51 EDT 2014. Contains 246886 sequences.