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Number of distinct squares that can be partitioned into distinct divisors of n.
2

%I #7 Nov 06 2022 07:49:02

%S 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,

%T 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,

%U 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

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

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

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

%F a(n) <= A078705(n).

%F a(A173494(n)) = 1.

%e divisors(9) = {1, 3, 9}: a(9) = #{1, 3+1, 9} = 3;

%e divisors(10) = {1, 2, 5, 10}: a(10) = #{1, 10+5+1} = 2;

%e divisors(12) = {1,2,3,4,6,12}: a(12) = #{1,4,9,16,25} = 5:

%e 2^2 = 4 = 3 + 1,

%e 3^2 = 6 + 3 = 6 + 2 + 1 = 4 + 3 + 2,

%e 4^2 = 12 + 4 = 12 + 3 + 1 = 6 + 4 + 3 + 2 + 1,

%e 5^2 = 12 + 6 + 4 + 3 = 12 + 6 + 4 + 2 + 1;

%e divisors(42)={1,2,3,6,7,14,21,42}: a(42)=#{k^2: 1<=k<=9}=9:

%e 2^2 = 3+1,

%e 3^2 = 7+2 = 6+3 = 6+2+1,

%e 4^2 = 14+2 = 7+6+3 = 7+6+2+1,

%e 5^2 = 21 + 3 + 1 = 14 + 7 + 3 + 1 = 14 + 6 + 3 + 2,

%e 6^2 = 21 + 14 + 1 = 21 + 7 + 6 + 2,

%e 7^2 = 42 + 7 = 42 + 6 + 1 = 21 + 14 + 7 + 6 + 1,

%e 8^2 = 42 + 21 + 1 = 42 + 14 + 7 + 1 = 42 + 14 + 6 + 2,

%e 9^2 = 42 + 21 + 14 + 3 + 1 = 42 + 21 + 7 + 6 + 3 + 2.

%Y Cf. A033630, A006532, A072243, A000203.

%K nonn

%O 1,3

%A _Reinhard Zumkeller_, Feb 20 2010