A005188 -id:A005188 - cp's OEIS frontend

A161949 Base-12 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 125, 811, 944, 1539, 28733, 193084, 887690, 2536330, 6884751, 17116683, 5145662993, 25022977605, 39989277598, 294245206529, 301149802206, 394317605931, 429649124722, 446779986586

Offset: 1

Joseph Myers, Jun 22 2009

Joseph Myers, Table of n, a(n) for n = 1..87 (the full list of terms, from Winter)
Henk Koppelaar, and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 12]^IntegerLength[#, 12]] &] (* Michael De Vlieger, Nov 04 2020 *)

A161953 Base-16 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 342, 371, 520, 584, 645, 1189, 1456, 1457, 1547, 1611, 2240, 2241, 2458, 2729, 2755, 3240, 3689, 3744, 3745, 47314, 79225, 177922, 177954, 368764, 369788, 786656, 786657, 787680, 787681, 811239, 812263, 819424, 819425, 820448, 820449, 909360

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 16|a(n) (n = 22, 26, 33, 41, 43, 47, 49, 51, 53, 61, 116, 149, 157, 196, 198, 204, 206, 243, 247), then a(n+1) = a(n) + 1. Zero also satisfies the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-16 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

			645 is in the sequence because 645 is 285 in hexadecimal and 2^3 + 8^3 + 5^3 = 645. (The exponent 3 is the number of hexadecimal digits.)

Joseph Myers, Table of n, a(n) for n=1..293 (the full list of terms, from Winter)
Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15).

Select[Range[10^7], # == Total[IntegerDigits[#, 16]^IntegerLength[#, 16]] &] (* Michael De Vlieger, Nov 04 2020 *)

isok(n) = {my(b=16, d=digits(n, b), e=#d); sum(k=1, #d, d[k]^e) == n;} \\ Michel Marcus, Feb 25 2019

select( is_A161953(n)={n==vecsum([d^#n|d<-n=digits(n,16)])}, [1..10^5]) \\ M. F. Hasler, Nov 22 2019

from itertools import islice, combinations_with_replacement
def A161953_gen(): # generator of terms
    for k in range(1,74):
        a = tuple(i**k for i in range(16))
        yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(int(d,16) for d in sorted(hex(x[0])[2:])) == x[1], \
                          ((sum(map(lambda y:a[y],b)),b) for b in combinations_with_replacement(range(16),k)))))
A161953_list = list(islice(A161953_gen(),30)) # Chai Wah Wu, Apr 21 2022

Terms sorted in increasing order by Pontus von Brömssen, Mar 03 2019

A046761 Largest number that is equal to sum of n-th powers of its digits.

Original entry on oeis.org

1, 9, 1, 407, 9474, 194979, 548834, 14459929, 88593477, 912985153, 4679307774, 94204591914, 1, 564240140138, 28116440335967, 1, 4338281769391371, 35875699062250035, 1, 4929273885928088826, 63105425988599693916

Offset: 0

David W. Wilson

Joseph Myers, Table of n, a(n) for n = 0..107

Cf. A023052, A003321, A046074, A005188.

In other bases: A162218 (base 3), A162221 (base 4), A162224 (base 5), A162227 (base 6), A162230 (base 7), A162233 (base 8), A162236 (base 9).

Cross-references edited by Joseph Myers, Jun 28 2009

A161948 Base-11 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 61, 72, 126, 370, 855, 1161, 1216, 1280, 10657, 16841, 16842, 17864, 17865, 36949, 36950, 63684, 66324, 71217, 90120, 99594, 99595, 141424, 157383, 1165098, 1165099, 5611015, 11959539, 46478562, 203821954, 210315331, 397800208, 826098079, 1308772162, 1399714480

Offset: 1

Joseph Myers, Jun 22 2009

From M. F. Hasler, Nov 20 2019: (Start)
Like the other single-digit terms, zero would satisfy the definition (n = Sum_{i=1..k} d[i]^k when d[1..k] are the base 11 digits of n), but here only positive numbers are considered.
Terms a(n+1) = a(n) + 1 (n = 20, 22, 24, 30, 34, 56, 67, 57, 195, ...) correspond to solutions a(n) that are multiples of 11, in which case a(n) + 1 is also a solution. (End)

			16841 = 11720_11 (= 1*11^4 + 1*11^3 + 7*11^2 + 2*11^1 + 0*11^0) = 1^5 + 1^5 + 7^5 + 2^5 + 0^5. It's easy to see that 16841 + 1 then also satisfies this relation, as for all terms that are multiples of 11. - _M. F. Hasler_, Nov 20 2019

Joseph Myers, Table of n, a(n) for n = 1..134 (the full list of terms, from Winter)
Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 11]^IntegerLength[#, 11]] &] (* Michael De Vlieger, Nov 04 2020 *)

select( {is_A161948(n)=n==vecsum([d^#n|d<-n=digits(n,11)])}, [0..10^5]) \\ This gives only terms < 10^5, for illustration of is_A161948(). - M. F. Hasler, Nov 20 2019

A161950 Base-13 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 45, 85, 98, 136, 160, 793, 794, 854, 1968, 8194, 62481, 167544, 167545, 294094, 320375, 323612, 325471, 325713, 350131, 365914, 2412003, 4861352, 21710514, 43757311, 43757312, 46299414, 51798568, 52994053

Offset: 1

Joseph Myers, Jun 22 2009

Joseph Myers, Table of n, a(n) for n = 1..201 (the full list of terms, from Winter)
Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 13]^IntegerLength[#, 13]] &] (* Michael De Vlieger, Nov 04 2020 *)

A161951 Base-14 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 244, 793, 282007, 10362564, 1445712420, 29546248981, 164159496751, 342515735622, 359057049845, 216210334578515, 324075236456868, 338527182572746, 338609726265795, 382789516519507, 435198066019184, 526088332647250

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 14|a(n) (n = 36, 46, 75, 77), then a(n+1) = a(n) + 1. Zero also satisfies the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-14 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

Joseph Myers, Table of n, a(n) for n=1..103 (the full list of terms, from Winter)
Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161952 (base 15), A161953 (base 16).

Select[Range[2 * 10^7], # == Total[IntegerDigits[#, 14]^IntegerLength[#, 14]] &] (* Michael De Vlieger, Nov 04 2020 *)

select( is_A161951(n)={n==vecsum([d^#n|d<-n=digits(n,14)])}, [1..10^6\3]) \\ M. F. Hasler, Nov 22 2019

A161952 Base-15 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 113, 128, 2755, 3052, 5059, 49074, 49089, 386862, 413951, 517902, 15219156, 18605333, 38009273, 40082196, 40310423, 40868227, 47527794, 100128060, 100128061, 100128188, 104189152, 105464820

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 15|a(n) (n = 32, 36, 40, 86, 100, 135, 143, 194, 197, 201), then a(n+1) = a(n) + 1. Zero also satisfies the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-15 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

Joseph Myers, Table of n, a(n) for n = 1..202 (the full list of terms, from Winter)
Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 15]^IntegerLength[#, 15]] &] (* Michael De Vlieger, Nov 04 2020 *)

select( is_A161952(n)={n==vecsum([d^#n|d<-n=digits(n,15)])}, [1..10^5]) \\ M. F. Hasler, Nov 22 2019

Terms sorted in increasing order by Pontus von Brömssen, Mar 03 2019

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

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, May 15 1998

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

			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.

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.

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.

Cf. A005188, A208130, A362843.

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

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 *)

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

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

A014576 Smallest n-digit narcissistic (or Armstrong) number: smallest n-digit number equal to sum of n-th powers of its digits (or 0 if no such number exists).

Original entry on oeis.org

1, 0, 153, 1634, 54748, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 0, 28116440335967, 0, 4338281769391370, 21897142587612075, 0, 1517841543307505039, 63105425988599693916, 128468643043731391252, 0

Offset: 1

N. J. A. Sloane

M. Gardner, The Magic Numbers of Dr Matrix. Prometheus, Buffalo, NY, 1985, p. 249.
C. A. Pickover, Keys to Infinity. New York: W. H. Freeman, pp. 169-170, 1995.

T. D. Noe, Table of n, a(n) for n=1..39 (complete sequence)
Harvey Heinz, Narcissistic Numbers
Mike Keith, Wild Narcissistic Numbers (Copy as of May 2008 on web.archive.org - page does not exist any more.)
Eric Weisstein's World of Mathematics, Narcissistic Number.

Cf. A001694, A007532, A005934, A005188, A003321, A023052, A046074.

(* This program is not suitable for more than 10 terms *) a[n_] := For[k = 10^(n-1), True, k++, If[k > 10^n - 1, Return[0], If[k == Total[ IntegerDigits[k]^IntegerLength[k] ], Return[k] ] ] ]; Table[ Print[an = a[n]]; an, {n, 1, 10}] (* Jean-François Alcover, Oct 15 2013 *)

Terms and links added by Patrick De Geest, Oct 1998

Broken links fixed by M. F. Hasler, Feb 12 2013

A256359 Numbers n such that there is at least one base b in which n is a multiple-digit narcissistic number.

Original entry on oeis.org

5, 8, 10, 13, 17, 18, 20, 25, 26, 28, 29, 32, 35, 37, 40, 41, 43, 45, 50, 52, 53, 55, 58, 61, 62, 65, 68, 72, 80, 82, 83, 85, 90, 92, 97, 98, 99, 101, 104, 109, 113, 117, 118, 122, 125, 126, 127, 128, 133, 134, 136, 145, 146, 148, 152, 153, 160, 162

Offset: 1

Tim Johannes Ohrtmann, Mar 26 2015

			a(1) = 5 because this is the first number that is a multiple-digit narcissistic number in at least one base (3).

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..53569
W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256360, A256361, A256362, A256363, A256364, A256365 (1 to 6 bases).
Cf. A256459 (first occurrences).

Select[Range@ 162, Function[k, AnyTrue[Range[2, k], Total[#^Length@ #] &@ IntegerDigits[k, #] == k &]]] (* Version 10, or *)
Select[Range@ 162, Function[k, Total@ Boole[Total[#^Length@ #] &@ IntegerDigits[k, #] == k & /@ Range[2, k]] > 0]] (* Michael De Vlieger, Apr 30 2016 *)

for(n=3,1000000, k=0; for(z=2,n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L,x,j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1,j, t+=L[p]^j); if(n==t, k++)); if(k>0, print1(n,", ")))

cp's OEIS Frontend

A161949 Base-12 Armstrong or narcissistic numbers (written in base 10).

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A161953 Base-16 Armstrong or narcissistic numbers (written in base 10).

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A046761 Largest number that is equal to sum of n-th powers of its digits.

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A161948 Base-11 Armstrong or narcissistic numbers (written in base 10).

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A161950 Base-13 Armstrong or narcissistic numbers (written in base 10).

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A161951 Base-14 Armstrong or narcissistic numbers (written in base 10).

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A161952 Base-15 Armstrong or narcissistic numbers (written in base 10).

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

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