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).
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
Offset: 1
Examples
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
References
- 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.
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)
- Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
- Hans Havermann, Cube wonders
- Douglas E. Iannucci and Bertrum Foster, Kaprekar Triples, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.8.
- Robert Munafo, Kaprekar Sequences
Crossrefs
Cf. A291461.
Programs
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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 *) -
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
Extensions
Entry revised by Larry Reeves (larryr(AT)acm.org), Apr 25 2001 and Dec 08 2002
Comments