A295753 The longest side of a minimal rectangle made from n squares.

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, 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, 9, 9, 10, 9, 9, 10, 9, 10, 10, 9, 10, 10, 9, 10, 10

Offset

1,2

Comments

A rectangle is minimal with respect to its longest side.

Links

Table of n, a(n) for n=1..83.

Formula

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

a(k^2) = k.

Example

The longest side of a minimal rectangle of n = 5 squares has side length a(5) = 4:
+---+---+---+---+    +---+---+---+---+
| 1 | 1 |       |    | 1 |       | 1 |
+---+---+   2   + or +---+   2   +---+
| 1 | 1 |       |    | 1 |       | 1 |
+---+---+---+---+    +---+---+---+---+
a(6) = 3:
+---+---+---+    +---+---+
| 1 | 1 | 1 |    | 1 | 1 |
+---+---+---+    +---+---+
| 1 |       | or | 1 | 1 |
+---+   2   +    +---+---+
| 1 |       |    | 1 | 1 |
+---+---+---+    +---+---+
a(7) = 4:
+---+---+---+---+
| 1 | 1 |       |
+---+---+   2   +
| 1 | 1 |       |
+---+---+---+---+
|       |       |
+   2   +   2   +
|       |       |
+---+---+---+---+

Mathematica

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 *)

Crossrefs

Sequence in context: A004596 A118653 A173753 * A144968 A239328 A076753

Adjacent sequences: A295750 A295751 A295752 * A295754 A295755 A295756

Keyword

nonn

Author

Peter Kagey, Nov 26 2017

Extensions

More terms from Giovanni Resta, Nov 28 2017

Status

approved