A006892 Representation as a sum of squares requires n squares with greedy algorithm. (Formerly M0860)

1, 2, 3, 7, 23, 167, 7223, 13053767, 42600227803223, 453694852221687377444001767, 51459754733114686962148583993443846186613037940783223

Offset

1,2

Comments

Of course Lagrange's theorem tells us that any positive integer can be written as a sum of at most four squares (cf. A004215).

Records in A053610. - Hugo van der Sanden, Jun 24 2015

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Rick L. Shepherd, Table of n, a(n) for n = 1..15

Art of Problem Solving, 2010 AMC 10A Problems/Problem 25 [From Rick L. Shepherd, Jan 28 2014]

E. Lemoine, Décomposition d'un nombre entier N en ses puissances nièmes maxima, C. R. Acad. Sci. Paris, Vol. 95, pp. 719-722, 1882 (then next pages).

E. Lemoine, Sur la décomposition d'un nombre en ses carrés maxima, Assoc. Française pour L'Avancement des Sciences (1896), 73-77.

Formula

For n >= 4, a(n) = a(n-1) + ((a(n-1)+1)/2)^2. - Joe K. Crump (joecr(AT)carolina.rr.com), Apr 16 2000

a(n) = n for n <= 3; for n > 3, a(n) = ((a(n-1)+3)/2)^2 - 2. - Arkadiusz Wesolowski, Mar 30 2013

a(n+2) = 2 * A053630(n) - 3. - Thomas Ordowski, Jul 14 2014

a(n+3) = A053630(n)^2 - 2. - Thomas Ordowski, Jul 19 2014

Example

Here is why a(5) = 23: start with 23, subtract largest square <= 23, which is 16, getting 7.
Now subtract largest square <= 7, which is 4, getting 3.
Now subtract largest square <= 3, which is 1, getting 2.
Now subtract largest square <= 2, which is 1, getting 1.
Now subtract largest square <= 1, which is 1, getting 0.
Thus 23 = 16+4+1+1+1.
It took 5 steps to get to 0, and 23 is the smallest number which takes 5 steps. - N. J. A. Sloane, Jan 29 2014

Prog

(PARI) a(n) = if (n <= 3, n , ((a(n-1)+3)/2)^2 - 2) \\ Michel Marcus, May 25 2013

Crossrefs

Cf. A004215, A053610.

Sequence in context: A108176 A111235 A066356 * A296397 A102710 A048824

Adjacent sequences: A006889 A006890 A006891 * A006893 A006894 A006895

Keyword

nonn

Author

Jeffrey Shallit

Extensions

Four more terms from Rick L. Shepherd, Jan 27 2014

Status

approved