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A337333 Number of pairs of odd divisors of n, (d1,d2), such that d1 <= d2.

%I A337333 #16 Dec 12 2021 22:53:13
%S A337333 1,1,3,1,3,3,3,1,6,3,3,3,3,3,10,1,3,6,3,3,10,3,3,3,6,3,10,3,3,10,3,1,
%T A337333 10,3,10,6,3,3,10,3,3,10,3,3,21,3,3,3,6,6,10,3,3,10,10,3,10,3,3,10,3,
%U A337333 3,21,1,10,10,3,3,10,10,3,6,3,3,21,3,10,10,3,3,15,3,3,10,10
%N A337333 Number of pairs of odd divisors of n, (d1,d2), such that d1 <= d2.
%C A337333 Number of distinct rectangles that can be made whose side lengths are odd divisors of n.
%H A337333 Antti Karttunen, <a href="/A337333/b337333.txt">Table of n, a(n) for n = 1..20000</a>
%F A337333 a(n) = Sum_{d1|n, d2|n, d1 and d2 odd, d1<=d2} 1.
%F A337333 From _Bernard Schott_, Aug 24 2020: (Start)
%F A337333 a(n) = 1 if and only if n = 2^k, k >= 0 (A000079).
%F A337333 a(n) = 3 if n is an odd prime. (End)
%F A337333 a(n) = A000217(A001227(n)). - _Antti Karttunen_, Dec 12 2021
%e A337333 a(15) = 10; There are 10 pairs of odd divisors of 15, (d1,d2), such that d1<=d2. They are: (1,1), (1,3), (1,5), (1,15), (3,3), (3,5), (3,15), (5,5), (5,15), (15,15). So a(15) = 10.
%e A337333 a(16) = 1; (1,1) is the only pair of odd divisors of 16, (d1,d2), such that d1<=d2. So a(16) = 1.
%e A337333 a(17) = 3; There are 3 pairs of odd divisors of 17, (d1,d2), such that d1<=d2. They are (1,1), (1,17) and (17,17). So a(17) = 3.
%e A337333 a(18) = 6; There are 6 pairs of odd divisors of 18, (d1,d2), such that d1<=d2. They are: (1,1), (1,3), (1,9), (3,3), (3,9) and (9,9). So a(18) = 6.
%t A337333 Table[Sum[Sum[Mod[i, 2]*Mod[k, 2] (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]
%o A337333 (PARI) A337333(n) = binomial(numdiv(n>>valuation(n,2))+1,2); \\ _Antti Karttunen_, Dec 12 2021
%Y A337333 Cf. A000079, A000217, A001227 (number of odd divisors), A335841.
%K A337333 nonn
%O A337333 1,3
%A A337333 _Wesley Ivan Hurt_, Aug 23 2020