reviewed

approved

proposed

reviewed

editing

proposed

a(n) <= A078705(n);

a(A173494(n)) = 1;

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

R. Reinhard Zumkeller, <a href="/A173493/b173493.txt">Table of n, a(n) for n = 1..250</a>

a(n) <= A078705(n).

a(A173494(n)) = 1.

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;

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.

Cf. A033630, A006532, A072243, A000203.

approved

editing

_Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), _, Feb 20 2010

R. Zumkeller, <a href="/A173493/b173493.txt">Table of n, a(n) for n = 1..250</a>

nonn,new

nonn

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

1,3

a(n) <= A078705(n);

a(A173494(n)) = 1;

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

R. Zumkeller, <a href="b173493.txt">Table of n, a(n) for n = 1..250</a>

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.

A033630, A006532, A072243, A000203.

nonn

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 20 2010

approved