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

%I A173493 #16 Apr 16 2025 04:36:08
%S A173493 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 A173493 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 A173493 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 A173493 Number of distinct squares that can be partitioned into distinct divisors of n.
%C A173493 The partitions of the squares are generally not unique, see examples.
%H A173493 Amiram Eldar, <a href="/A173493/b173493.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..250 from Reinhard Zumkeller)
%F A173493 a(n) <= A078705(n).
%F A173493 a(A173494(n)) = 1.
%e A173493 divisors(9) = {1, 3, 9}: a(9) = #{1, 3+1, 9} = 3.
%e A173493 divisors(10) = {1, 2, 5, 10}: a(10) = #{1, 10+5+1} = 2.
%e A173493 divisors(12) = {1,2,3,4,6,12}: a(12) = #{1,4,9,16,25} = 5:
%e A173493   2^2 = 4 = 3 + 1,
%e A173493   3^2 = 6 + 3 = 6 + 2 + 1 = 4 + 3 + 2,
%e A173493   4^2 = 12 + 4 = 12 + 3 + 1 = 6 + 4 + 3 + 2 + 1,
%e A173493   5^2 = 12 + 6 + 4 + 3 = 12 + 6 + 4 + 2 + 1.
%e A173493 divisors(42)={1,2,3,6,7,14,21,42}: a(42)=#{k^2: 1<=k<=9}=9:
%e A173493   2^2 = 3+1,
%e A173493   3^2 = 7+2 = 6+3 = 6+2+1,
%e A173493   4^2 = 14+2 = 7+6+3 = 7+6+2+1,
%e A173493   5^2 = 21 + 3 + 1 = 14 + 7 + 3 + 1 = 14 + 6 + 3 + 2,
%e A173493   6^2 = 21 + 14 + 1 = 21 + 7 + 6 + 2,
%e A173493   7^2 = 42 + 7 = 42 + 6 + 1 = 21 + 14 + 7 + 6 + 1,
%e A173493   8^2 = 42 + 21 + 1 = 42 + 14 + 7 + 1 = 42 + 14 + 6 + 2,
%e A173493   9^2 = 42 + 21 + 14 + 3 + 1 = 42 + 21 + 7 + 6 + 3 + 2.
%t A173493 a[n_] := Module[{d = Divisors[n], sum, sq, x}, sum = Plus @@ d; sq = Range[Floor[Sqrt[sum]]]^2; Count[CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sq]], _?(# > 0&)]]; Array[a, 100] (* _Amiram Eldar_, Apr 16 2025 *)
%Y A173493 Cf. A000203, A006532, A033630, A072243, A078705, A173494.
%K A173493 nonn
%O A173493 1,3
%A A173493 _Reinhard Zumkeller_, Feb 20 2010