A173493 Number of distinct squares that can be partitioned into distinct divisors of n.

1, 1, 2, 2, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 4, 5, 1, 6, 1, 6, 3, 3, 1, 7, 2, 2, 4, 7, 1, 8, 1, 7, 3, 2, 2, 9, 1, 1, 3, 9, 1, 9, 1, 7, 7, 3, 1, 11, 2, 5, 2, 4, 1, 10, 2, 10, 2, 1, 1, 12, 1, 2, 7, 11, 1, 12, 1, 4, 2, 11, 1, 13, 1, 1, 9, 7, 1, 12, 1, 13, 6, 1, 1, 14, 1, 1, 2, 13, 1, 15, 1, 6, 2, 3, 3, 15, 1, 8

Offset

1,3

Comments

The partitions of the squares are generally not unique, see examples.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Formula

a(n) <= A078705(n).

a(A173494(n)) = 1.

Example

divisors(9) = {1, 3, 9}: a(9) = #{1, 3+1, 9} = 3;
divisors(10) = {1, 2, 5, 10}: a(10) = #{1, 10+5+1} = 2;
divisors(12) = {1,2,3,4,6,12}: a(12) = #{1,4,9,16,25} = 5:
  2^2 = 4 = 3 + 1,
  3^2 = 6 + 3 = 6 + 2 + 1 = 4 + 3 + 2,
  4^2 = 12 + 4 = 12 + 3 + 1 = 6 + 4 + 3 + 2 + 1,
  5^2 = 12 + 6 + 4 + 3 = 12 + 6 + 4 + 2 + 1;
divisors(42)={1,2,3,6,7,14,21,42}: a(42)=#{k^2: 1<=k<=9}=9:
  2^2 = 3+1,
  3^2 = 7+2 = 6+3 = 6+2+1,
  4^2 = 14+2 = 7+6+3 = 7+6+2+1,
  5^2 = 21 + 3 + 1 = 14 + 7 + 3 + 1 = 14 + 6 + 3 + 2,
  6^2 = 21 + 14 + 1 = 21 + 7 + 6 + 2,
  7^2 = 42 + 7 = 42 + 6 + 1 = 21 + 14 + 7 + 6 + 1,
  8^2 = 42 + 21 + 1 = 42 + 14 + 7 + 1 = 42 + 14 + 6 + 2,
  9^2 = 42 + 21 + 14 + 3 + 1 = 42 + 21 + 7 + 6 + 3 + 2.

Crossrefs

Cf. A033630, A006532, A072243, A000203.

Sequence in context: A277730 A174167 A159876 * A044924 A057036 A069004

Adjacent sequences: A173490 A173491 A173492 * A173494 A173495 A173496

Keyword

nonn

Author

Reinhard Zumkeller, Feb 20 2010

Status

approved