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A032799
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Numbers n such that n equals the sum of its digits raised to the consecutive powers (1,2,3,...).
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8
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, 12157692622039623539
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OFFSET
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1,3
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COMMENTS
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Lemma: The sequence is finite with all terms in the sequence having at most 22 digits. Proof: Let n be an m-digit natural number in the sequence for some m. Then 10^(m-1)<=n and n<=9+9^2+...9^m = 9(9^m-1)/8<(9^(m+1))/8. Thus 10^(m-1)<(9^(m+1))/8. Taking logarithms of both sides and solving yields m<22.97 QED. Note proof is identical to that for A208130. [Francis J. McDonnell, Apr 14 2012]
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 175, p. 55, Ellipses, Paris 2008.
Ken Follett, Code to Zero, Dutton, a Penguin Group, NY 2000, p. 84.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, London, 1986, Entry 175.
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LINKS
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Table of n, a(n) for n=1..20.
Eric Weisstein's World of Mathematics, Narcissistic Number
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EXAMPLE
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89 = 8^1 + 9^2.
175 = 1^1 + 7^2 + 5^3.
2427 = 2^1 + 4^2 + 2^3 + 7^4.
2646798 = 2^1 + 6^2 + 4^3 + 6^4 + 7^5 + 9^6 + 8^7.
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MATHEMATICA
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f[n_] := Plus @@ (IntegerDigits[n]^Range[ Floor[ Log[10, n] + 1]]); Select[ Range[10^7], f[ # ] == # &] (* from Robert G. Wilson v, May 04 2005 *)
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CROSSREFS
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Sequence in context: A023106 A135480 A098766 * A208130 A160343 A024664
Adjacent sequences: A032796 A032797 A032798 * A032800 A032801 A032802
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KEYWORD
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nonn,base,fini,full,nice
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AUTHOR
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Patrick De Geest, May 15, 1998.
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EXTENSIONS
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Corrected by Macsy Zhang (macsy(AT)21cn.com), Feb 17, 2002
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STATUS
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approved
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