(Translated by https://www.hiragana.jp/)
# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a005188 Showing 1-1 of 1 %I A005188 M0488 #145 Mar 09 2024 04:42:58 %S A005188 1,2,3,4,5,6,7,8,9,153,370,371,407,1634,8208,9474,54748,92727,93084, %T A005188 548834,1741725,4210818,9800817,9926315,24678050,24678051,88593477, %U A005188 146511208,472335975,534494836,912985153,4679307774,32164049650,32164049651 %N 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. %C A005188 A finite sequence, the 88th and last term being 115132219018763992565095597973971522401. %C A005188 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 A005188 These are the fixed points in the "Recurring Digital Invariant Variant" described in A151543. %C A005188 a(15) = A229381(3) = 8208 is the "Simpsons' narcissistic number". %C A005188 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 A005188 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 A005188 Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 88, pp. 30-31, Ellipses, Paris 2008. %D A005188 Lionel E. Deimel, Jr. and Michael T. Jones, Finding Pluperfect Digital Invariants: Techniques, Results and Observations, J. Rec. Math., 14 (1981), 87-108. %D A005188 Jean-Pierre Lamoitier, Fifty Basic Exercises. SYBEX Inc., 1981. %D A005188 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68. %D A005188 Alfred S. Posamentier, Numbers: Their Tales, Types, and Treasures, Prometheus Books, 2015, pp. 242-244. %D A005188 Joe Roberts, The Lure of the Integers, The Mathematical Association of America, 1992, page 36. %D A005188 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A005188 T. D. Noe, Table of n, a(n) for n = 1..88 (the full list of terms, from Winter) %H A005188 Anonymous, Narcissistic number. %H A005188 Michael F. Armstrong, A Brief Introduction to Armstrong Numbers. %H A005188 Pat Ballew, The Cubic Attractiveness of 153, Pat's Blog, May 30, 2023. %H A005188 Hans J. de Jong, Letter to N. J. A. Sloane, Mar 8 1988. %H A005188 Lionel E. Deimel, Armstrong Numbers. %H A005188 Lionel E. Deimel, Mystery Solved!, Lionel Deimel’s Web Log, May 5, 2010. %H A005188 Lionel E. Deimel, Narcissistic Numbers. %H A005188 Martin Gardner & N. J. A. Sloane, Correspondence, 1973-74. %H A005188 Harvey Heinz, Narcissistic Numbers (backup from March 2018 on web/archive.org: page no longer available), Sep. 1998, last updated in Sep. 2010. %H A005188 History of Science and Mathematics StackExchange, Armstrong numbers - who is or was Armstrong?, 2021. %H A005188 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. %H A005188 Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224. %H A005188 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. %H A005188 Walter Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs) %H A005188 B. Shader, Armstrong number. %H A005188 Eric Weisstein's World of Mathematics, Narcissistic Number. %H A005188 Wikipedia, Narcissistic number. %H A005188 Robert G. Wilson v, Letter to N. J. A. Sloane, Jan 23 1989. %H A005188 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. %e A005188 153 = 1^3 + 5^3 + 3^3, %e A005188 8208 = 8^4 + 2^4 + 0^4 + 8^4, %e A005188 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7. %e A005188 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 A005188 filter:= proc(k) local d; %p A005188 d:= 1 + ilog10(k); %p A005188 add(s^d, s=convert(k,base,10)) = k %p A005188 end proc: %p A005188 select(filter, [$1..10^6]); # _Robert Israel_, Jan 02 2015 %t A005188 f[n_] := Plus @@ (IntegerDigits[n]^Floor[ Log[10, n] + 1]); Select[ Range[10^7], f[ # ] == # &] (* _Robert G. Wilson v_, May 04 2005 *) %t A005188 Select[Range[10^7],#==Total[IntegerDigits[#]^IntegerLength[#]]&] (* _Harvey P. Dale_, Sep 30 2011 *) %o A005188 (PARI) is(n)=my(v=digits(n));sum(i=1,#v,v[i]^#v)==n \\ _Charles R Greathouse IV_, Nov 20 2012 %o A005188 (PARI) select( is_A005188(n)={n==vecsum([d^#n|d<-n=digits(n)])}, [0..9999]) \\ _M. F. Hasler_, Nov 18 2019 %o A005188 (Python) %o A005188 from itertools import combinations_with_replacement %o A005188 A005188_list = [] %o A005188 for k in range(1,10): %o A005188 a = [i**k for i in range(10)] %o A005188 for b in combinations_with_replacement(range(10),k): %o A005188 x = sum(map(lambda y:a[y],b)) %o A005188 if x > 0 and tuple(int(d) for d in sorted(str(x))) == b: %o A005188 A005188_list.append(x) %o A005188 A005188_list = sorted(A005188_list) # _Chai Wah Wu_, Aug 25 2015 %Y A005188 Cf. A001694, A007532, A005934, A003321, A014576, A046074. %Y A005188 Similar to but different from A023052. %Y A005188 Cf. A151543. %Y A005188 Cf. A010343 to A010354 (bases 4 to 9). - _R. J. Mathar_, Jun 28 2009 %K A005188 nonn,base,fini,full,nice %O A005188 1,2 %A A005188 _N. J. A. Sloane_, _Robert G. Wilson v_ %E A005188 32164049651 from Amit Munje (amit.munje(AT)gmail.com), Oct 07 2006 %E A005188 In order to agree with the Definition, first comment modified by _Jonathan Sondow_, Jan 02 2015 %E A005188 Comment in name moved to comment section and links edited by _M. F. Hasler_, Oct 18 2018 %E A005188 "Positive" added to definition by _N. J. A. Sloane_, Nov 18 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE