A032799 Numbers k such that k equals the sum of its digits raised to the consecutive powers (1,2,3,...).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, 12157692622039623539

Offset: 1

Author

Patrick De Geest, May 15 1998

Keywords

• nonn
• base
• fini
• full
• nice

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

Examples

			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.
		

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.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 37.
• 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.
• 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, Narcissistic Number.

Crossrefs

Cf. A005188, A208130, A362843.

Programs

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 *)

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

Extensions

Corrected by Macsy Zhang (macsy(AT)21cn.com), Feb 17 2002