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A173493
Number of distinct squares that can be partitioned into distinct divisors of n.
2
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
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 20 2010
STATUS
approved