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Search: id:A005188
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| A005188 |
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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)
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+0
41
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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.
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REFERENCES
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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).
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LINKS
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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, Link to a section of The World of Mathematics (1).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).
Wikipedia, Narcissistic number
D. T. Winter, Table of Armstrong Numbers
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EXAMPLE
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153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.
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MATHEMATICA
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f[n_] := Plus @@ (IntegerDigits[n]^Floor[ Log[10, n] + 1]); Select[ Range[10^7], f[ # ] == # &] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 04 2005)
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CROSSREFS
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Cf. A001694, A007532, A005934, A003321, A014576, A046074.
Similar to but different from A023052.
Cf. A151543.
Cf. A010343 to A010354 (bases 4 to 9). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 28 2009]
Sequence in context: A151544 A032561 A023052 this_sequence A032569 A039723 A002998
Adjacent sequences: A005185 A005186 A005187 this_sequence A005189 A005190 A005191
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KEYWORD
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nonn,base,fini,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
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32164049651 from Amit Munje (amit.munje(AT)gmail.com), Oct 07 2006
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