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A005188 Armstrong (or Plus Perfect, or narcissistic) numbers: n-digit numbers equal to sum of n-th powers of their digits (a finite sequence, the last term being 115132219018763992565095597973971522401).
(Formerly M0488)
50
1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774, 32164049650, 32164049651 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let n = d_1 d_2 ... d_k in base 10; then n = Sum_{i=1..k} d_i^k.

These are the fixed points in the "Recurring Digital Invariant Variant" described in A151543.

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 88, pp 30-31, Ellipses, Paris 2008.

Lionel E. Deimel, Jr. and Michael T. Jones, Finding Pluperfect Digital Invariants: Techniques, Results and Observations, J. Rec. Math., 14 (1981), 87-108.

J. P. Lamoitier, Fifty Basic Exercises. SYBEX Inc., 1981.

Tomas Antonio Mendes Oliveira e Silva (tos(AT)ci.ua.pt) gave the full sequence in a posting (Article 42889) to sci.math on May 09 1994.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.

Joe Roberts, "The Lure of the Integers", page 35.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..88 (the full list of terms, from Winter)

Anonymous, Narcissistic number

L. E. Deimel, Narcissistic Numbers

H. Heinz, Narcissistic Numbers

W. Lopez, PlanetMath.Org, Armstrong number

W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)

B. Shader, Armstrong number

Eric Weisstein's World of Mathematics, Armstrong Number.

Eric Weisstein's World of Mathematics, Narcissistic Number

Wikipedia, Narcissistic number

D. T. Winter, Table of Armstrong Numbers

FORMULA

Let f(x)=Sum_{n=0,floor(log(x))} (floor(x/10^n)-10*floor(x/10^(n + 1)))^(floor(log(x))+1)- x. If f(x)=0, then x is a narcissistic number. - José de Jesús Camacho Medina, Mar 15 2014

EXAMPLE

153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

8208 = 8^4 + 2^4 + 0^4 + 8^4.

MATHEMATICA

f[n_] := Plus @@ (IntegerDigits[n]^Floor[ Log[10, n] + 1]); Select[ Range[10^7], f[ # ] == # &] (* Robert G. Wilson v, May 04 2005 *)

Select[Range[10^7], #==Total[IntegerDigits[#]^IntegerLength[#]]&] (* Harvey P. Dale, Sep 30 2011 *)

s[x_]:= Sum[(Floor[x/10^n]-10*Floor[x/10^(n + 1)])^(Floor[Log[10, x]]+1), {n, 0, Floor[Log[10, x]] }]; Select[Range[999], s[#] == # &] (*José de Jesús Camacho Medina, Mar 15 2014 *)

PROG

(PARI) is(n)=my(v=digits(n)); sum(i=1, #v, v[i]^#v)==n \\ Charles R Greathouse IV, Nov 20 2012

CROSSREFS

Cf. A001694, A007532, A005934, A003321, A014576, A046074.

Similar to but different from A023052.

Cf. A151543.

Cf. A010343 to A010354 (bases 4 to 9). - R. J. Mathar, Jun 28 2009

Sequence in context: A151544 A032561 A023052 * A032569 A039723 A002998

Adjacent sequences:  A005185 A005186 A005187 * A005189 A005190 A005191

KEYWORD

nonn,base,fini,full,nice

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

32164049651 from Amit Munje (amit.munje(AT)gmail.com), Oct 07 2006

STATUS

approved

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Last modified December 19 19:48 EST 2014. Contains 252239 sequences.