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A005188 Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: m-digit numbers equal to sum of the m-th powers of their digits.
(Formerly M0488)
75
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

A finite sequence, the 88th and last term being 115132219018763992565095597973971522401.

Let k = d_1 d_2 ... d_n in base 10; then k is in the sequence iff k = Sum_{i=1..n} d_i^n.

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

a(15) = A229381(3) = 8208 is the "Simpsons' narcissistic number".

If a(n) is a multiple of 10, then a(n+1) = a(n) + 1, and if a(n) == 1 (mod 10) then a(n-1) = a(n) - 1 except for n = 1, cf. Examples. - M. F. Hasler, Oct 18 2018

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

H. de Jong, Letter to N. J. A. Sloane, Mar 8 1988.

L. E. Deimel, Narcissistic Numbers

M. Gardner & N. J. A. Sloane, Correspondence, 1973-74

H. Heinz, Narcissistic Numbers (backup from March 2018 on web/archive.org: page no more available), Sep. 1998, last updated in Sep. 2010

L. H. & W. Lopez, PlanetMath.Org, Armstrong number (latest backup on web.archive.org of ArmstrongNumber.html from 2012), published by L.H. not later than July 2007.

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

B. Shader, Armstrong number

Eric Weisstein's World of Mathematics, Narcissistic Number

Wikipedia, Narcissistic number

Robert G. Wilson v, Letter to N. J. A. Sloane, Jan 23 1989.

D. T. Winter, Table of Armstrong Numbers (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

EXAMPLE

153 = 1^3 + 5^3 + 3^3,

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

4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

The eight terms 370, 24678050, 32164049650, 4338281769391370, 3706907995955475988644380, 19008174136254279995012734740, 186709961001538790100634132976990 and 115132219018763992565095597973971522400 end in a digit zero, therefore their successor a(n) + 1 is the next term a(n+1). This also yields the last term of the sequence. The initial a(1) = 1 is the only term ending in a digit 1 not preceded by a(n) - 1. - M. F. Hasler, Oct 18 2018

MAPLE

filter:= proc(k) local d;

d:= 1 + ilog10(k);

add(s^d, s=convert(k, base, 10)) = k

end proc:

select(filter, [$1..10^6]); # Robert Israel, Jan 02 2015

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 *)

PROG

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

(Python)

from itertools import combinations_with_replacement

A005188_list = []

for k in range(1, 10):

    a = [i**k for i in range(10)]

    for b in combinations_with_replacement(range(10), k):

        x = sum(map(lambda y:a[y], b))

        if x > 0 and tuple(int(d) for d in sorted(str(x))) == b:

            A005188_list.append(x)

A005188_list = sorted(A005188_list) # Chai Wah Wu, Aug 25 2015

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 A254960 A039723

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

In order to agree with the Definition, first comment modified by Jonathan Sondow, Jan 02 2015

Comment in name moved to comment section and links edited by M. F. Hasler, Oct 18 2018

STATUS

approved

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Last modified November 14 23:06 EST 2018. Contains 317221 sequences. (Running on oeis4.)