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A006887
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Kaprekar triples: q such that q = x + y + z and q^3 = x*10^2n + y*10^n + z, where z < 10^n and n is the number of digits in q. q is not a power of 10 (except q=1).
(Formerly M4478)
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9
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1, 8, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, 7172, 27100, 44443, 55556, 60434, 77778, 143857, 208494, 226071, 279720, 313390, 324675, 329967, 346060, 368928, 395604, 422577, 427868, 461539, 472823, 478115, 488214, 494208
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OFFSET
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1,2
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COMMENTS
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The initial term a(1) = 1 is somewhat conventional: it is the only term with x = y = 0 and q = z = 10^k, which is explicitly allowed only for k = 0 and forbidden for k > 0. In all other cases, 0 < x, y, z < q, and q^3 has the same number of digits as x*10^2n. - M. F. Hasler, Aug 24 2017
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.
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LINKS
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Douglas E. Iannucci and Bertrum Foster, Kaprekar Triples, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.8.
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EXAMPLE
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1 = 0 + 0 + 1 and 1^3 = (00)1 (cf. comment),
8 = 5 + 1 + 2 and 8^3 = 512,
45 = 9 + 11 + 25, and 45^3 = 91125. - M. F. Hasler, Aug 24 2017
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MATHEMATICA
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ok[n_] := n==1 || Block[{k = 10^IntegerLength[n], m = n^3}, n == Mod[m, k] + Floor[ m/k^2] + Mod[Floor[m/k], k] && ! IntegerQ@ Log10@ n]; Select[ Range@ 500000, ok] (* Giovanni Resta, Aug 23 2017 *)
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PROG
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(PARI) m=1; for(n=1, 6, for(q=m+(n>1), -1+m*=10, q==sumdigits(q^3, m)&&print1(q", "))) \\ M. F. Hasler, Aug 24 2017
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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Entry revised by Larry Reeves (larryr(AT)acm.org), Apr 25 2001 and Dec 08 2002
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STATUS
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approved
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