

A006887


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)


9



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

1,2


COMMENTS

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


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 151.


LINKS

Jack Brennen and Hans Havermann, Table of n, a(n) for n = 1..1000 (First 200 terms from Giovanni Resta.)
Futility Closet's "Math Notes", Shows the cubes of a(9) to a(13)
Hans Havermann, Cube wonders
Douglas E. Iannucci and Bertrum Foster, Kaprekar Triples, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.8.
R. Munafo, Kaprekar Sequences


EXAMPLE

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


MATHEMATICA

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 *)


PROG

(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


CROSSREFS

Cf. A291461.
Sequence in context: A055422 A204618 A289896 * A009369 A120044 A328885
Adjacent sequences: A006884 A006885 A006886 * A006888 A006889 A006890


KEYWORD

nonn,base


AUTHOR

Robert Munafo


EXTENSIONS

Entry revised by Larry Reeves (larryr(AT)acm.org), Apr 25 2001 and Dec 08 2002


STATUS

approved



