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

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

Offset: 1

Author

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

Keywords

• nonn
• base
• fini
• full
• nice

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

Named after Michael Frederick Armstrong (1941-2020), who used these numbers in his computing class at the University of Rochester in the mid 1960's. - Amiram Eldar, Mar 09 2024

Examples

			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
		

References

• Jean-Marie 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.
• Jean-Pierre Lamoitier, Fifty Basic Exercises. SYBEX Inc., 1981.
• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
• Alfred S. Posamentier, Numbers: Their Tales, Types, and Treasures, Prometheus Books, 2015, pp. 242-244.
• Joe Roberts, The Lure of the Integers, The Mathematical Association of America, 1992, page 36.
• 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.
• Michael F. Armstrong, A Brief Introduction to Armstrong Numbers.
• Pat Ballew, The Cubic Attractiveness of 153, Pat's Blog, May 30, 2023.
• Hans J. de Jong, Letter to N. J. A. Sloane, Mar 8 1988.
• Lionel E. Deimel, Armstrong Numbers.
• Lionel E. Deimel, Mystery Solved!, Lionel Deimel's Web Log, May 5, 2010.
• Lionel E. Deimel, Narcissistic Numbers.
• Martin Gardner & N. J. A. Sloane, Correspondence, 1973-74.
• Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
• Harvey Heinz, Narcissistic Numbers (backup from March 2018 on web/archive.org: page no longer available), Sep. 1998, last updated in Sep. 2010.
• History of Science and Mathematics StackExchange, Armstrong numbers - who is or was Armstrong?, 2021.
• 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.
• Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224.
• Tomas Antonio Mendes Oliveira e Silva (tos(AT)ci.ua.pt), Loneliness of the Factorions, gave the full sequence in a posting (Article 42889) to sci.math on May 09 1994.
• Walter 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.

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

Programs

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 *)

PARI

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

PARI

select( is_A005188(n)={n==vecsum([d^#n|d<-n=digits(n)])}, [0..9999]) \\ M. F. Hasler, Nov 18 2019

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

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

"Positive" added to definition by N. J. A. Sloane, Nov 18 2019