A295753 - OEIS

%I #14 Nov 29 2017 04:14:13

%S 1,2,3,2,4,3,4,4,3,4,5,4,4,5,5,4,5,6,5,5,6,5,7,6,5,7,6,6,7,6,7,7,6,7,

%T 7,6,7,7,7,7,7,7,7,8,8,7,8,8,7,8,9,8,8,9,8,8,9,8,9,9,8,9,9,8,9,9,9,10,

%U 9,9,10,9,9,10,9,10,10,9,10,10,9,10,10

%N The longest side of a minimal rectangle made from n squares.

%C A rectangle is minimal with respect to its longest side.

%F a(k^2 - 1) = k + 1.

%F a(k^2) = k.

%e The longest side of a minimal rectangle of n = 5 squares has side length a(5) = 4:

%e +---+---+---+---+    +---+---+---+---+

%e | 1 | 1 |       |    | 1 |       | 1 |

%e +---+---+   2   + or +---+   2   +---+

%e | 1 | 1 |       |    | 1 |       | 1 |

%e +---+---+---+---+    +---+---+---+---+

%e a(6) = 3:

%e +---+---+---+    +---+---+

%e | 1 | 1 | 1 |    | 1 | 1 |

%e +---+---+---+    +---+---+

%e | 1 |       | or | 1 | 1 |

%e +---+   2   +    +---+---+

%e | 1 |       |    | 1 | 1 |

%e +---+---+---+    +---+---+

%e a(7) = 4:

%e +---+---+---+---+

%e | 1 | 1 |       |

%e +---+---+   2   +

%e | 1 | 1 |       |

%e +---+---+---+---+

%e |       |       |

%e +   2   +   2   +

%e |       |       |

%e +---+---+---+---+

%t good[s_, w_] := w == {} || s >= Ceiling[Sqrt[Length[w]]] Max[w]; no1[w_] := Select[w, # > 1 &]; a[n_] := Block[{m = Ceiling@ Sqrt@ n, x, p, ok = False}, While[! ok, Do[p = Select[Sqrt@ IntegerPartitions[m x, {n}, Range[Sqrt@ n]^2], good[x, no1[#]] &]; If[p != {}, ok = True; Print["a[", n, "]=", m, "  (", x, " x ", m, ") = ", no1@ First@ p]; ok = True; Break[]], {x, m}]; m++]; m - 1]; Array[a, 100] (* correct at least for n <= 1000, _Giovanni Resta_, Nov 28 2017 *)

%K nonn

%O 1,2

%A _Peter Kagey_, Nov 26 2017

%E More terms from _Giovanni Resta_, Nov 28 2017