%I M0488 #148 May 31 2024 22:07:43

%S 1,2,3,4,5,6,7,8,9,153,370,371,407,1634,8208,9474,54748,92727,93084,

%T 548834,1741725,4210818,9800817,9926315,24678050,24678051,88593477,

%U 146511208,472335975,534494836,912985153,4679307774,32164049650,32164049651

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

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

%C 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.

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

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

%C 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

%C 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

%D Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 88, pp. 30-31, Ellipses, Paris 2008.

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

%D Jean-Pierre Lamoitier, Fifty Basic Exercises. SYBEX Inc., 1981.

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

%D Alfred S. Posamentier, Numbers: Their Tales, Types, and Treasures, Prometheus Books, 2015, pp. 242-244.

%D Joe Roberts, The Lure of the Integers, The Mathematical Association of America, 1992, page 36.

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

%H T. D. Noe, <a href="/A005188/b005188.txt">Table of n, a(n) for n = 1..88</a> (the full list of terms, from Winter)

%H Anonymous, <a href="http://everything2.net/index.pl?node_id=1525466&amp;displaytype=printable&amp;lastnode_id=1525466">Narcissistic number</a>.

%H Michael F. Armstrong, <a href="https://www.deimel.org/rec_math/armstrong.pdf">A Brief Introduction to Armstrong Numbers</a>.

%H Pat Ballew, <a href="https://pballew.blogspot.com/2023/05/the-cubic-attractiveness-of-153.html">The Cubic Attractiveness of 153</a>, Pat's Blog, May 30, 2023.

%H Hans J. de Jong, <a href="/A005188/a005188_1.pdf">Letter to N. J. A. Sloane</a>, Mar 8 1988.

%H Lionel E. Deimel, <a href="https://www.deimel.org/rec_math/DI_6.htm">Armstrong Numbers</a>.

%H Lionel E. Deimel, <a href="http://blog.deimel.org/2010/05/mystery-solved.html">Mystery Solved!</a>, Lionel Deimel’s Web Log, May 5, 2010.

%H Lionel E. Deimel, <a href="http://www.deimel.org/rec_math/DI_1.htm">Narcissistic Numbers</a>.

%H Martin Gardner & N. J. A. Sloane, <a href="/A003154/a003154.pdf">Correspondence, 1973-74</a>.

%H Harvey Heinz, <a href="http://web.archive.org/web/20180303194332/http://www.magic-squares.net:80/narciss.htm#Perfect%20Digital%20Invariants">Narcissistic Numbers</a> (backup from March 2018 on web/archive.org: page no longer available), Sep. 1998, last updated in Sep. 2010.

%H History of Science and Mathematics StackExchange, <a href="https://hsm.stackexchange.com/questions/13913/armstrong-numbers-who-is-or-was-armstrong">Armstrong numbers - who is or was Armstrong?</a>, 2021.

%H L. H. & W. Lopez, PlanetMath.Org, <a href="http://web.archive.org/web/20121109214834/http://planetmath.org:80/encyclopedia/ArmstrongNumber.html">Armstrong number</a> (latest backup on web.archive.org of ArmstrongNumber.html from 2012), published by L.H. not later than July 2007.

%H Gordon L. Miller and Mary T. Whalen, <a href="https://www.fq.math.ca/Scanned/30-3/miller.pdf">Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3</a>, Fibonacci Quarterly, 30-3 (1992), 221-224.

%H Tomas Antonio Mendes Oliveira e Silva (tos(AT)ci.ua.pt), <a href="https://groups.google.com/g/sci.math/c/MVKrv8NsYr8/m/BWzwpfF5rAsJ">Loneliness of the Factorions</a>, gave the full sequence in a posting (Article 42889) to sci.math on May 09 1994.

%H Walter Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/pdi.html">Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)</a>

%H B. Shader, <a href="http://everything2.net/index.pl?node_id=1407017&amp;displaytype=printable&amp;lastnode_id=1407017">Armstrong number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NarcissisticNumber.html">Narcissistic Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic number</a>.

%H Robert G. Wilson v, <a href="/A005188/a005188.pdf">Letter to N. J. A. Sloane</a>, Jan 23 1989.

%H D. T. Winter, <a href="http://web.archive.org/web/20100109234250/http://ftp.cwi.nl:80/dik/Armstrong">Table of Armstrong Numbers</a> (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

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

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

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

%e 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

%p filter:= proc(k) local d;

%p d:= 1 + ilog10(k);

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

%p end proc:

%p select(filter, [$1..10^6]); # _Robert Israel_, Jan 02 2015

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

%t Select[Range[10^7],#==Total[IntegerDigits[#]^IntegerLength[#]]&] (* _Harvey P. Dale_, Sep 30 2011 *)

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

%o (PARI) select( is_A005188(n)={n==vecsum([d^#n|d<-n=digits(n)])}, [0..9999]) \\ _M. F. Hasler_, Nov 18 2019

%o (Python)

%o from itertools import combinations_with_replacement

%o A005188_list = []

%o for k in range(1,10):

%o a = [i**k for i in range(10)]

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

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

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

%o A005188_list.append(x)

%o A005188_list = sorted(A005188_list) # _Chai Wah Wu_, Aug 25 2015

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

%Y Similar to but different from A023052.

%Y Cf. A151543.

%Y Cf. A010343 to A010354 (bases 4 to 9). - _R. J. Mathar_, Jun 28 2009

%K nonn,base,fini,full,nice

%O 1,2

%A _N. J. A. Sloane_, _Robert G. Wilson v_

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

%E In order to agree with the Definition, first comment modified by _Jonathan Sondow_, Jan 02 2015

%E Comment in name moved to comment section and links edited by _M. F. Hasler_, Oct 18 2018

%E "Positive" added to definition by _N. J. A. Sloane_, Nov 18 2019