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A337175 Number of pairs of divisors of n, (d1,d2), such that d1 <= d2 and d1*d2 < n.

%I A337175 #42 May 30 2025 10:16:20
%S A337175 0,1,1,2,1,4,1,4,2,4,1,9,1,4,4,6,1,9,1,9,4,4,1,16,2,4,4,9,1,16,1,9,4,
%T A337175 4,4,20,1,4,4,16,1,16,1,9,9,4,1,25,2,9,4,9,1,16,4,16,4,4,1,36,1,4,9,
%U A337175 12,4,16,1,9,4,16,1,36,1,4,9,9,4,16,1,25,6,4,1,36,4,4
%N A337175 Number of pairs of divisors of n, (d1,d2), such that d1 <= d2 and d1*d2 < n.
%H A337175 Amiram Eldar, <a href="/A337175/b337175.txt">Table of n, a(n) for n = 1..10000</a>
%F A337175 a(n) = Sum_{d1|n, d2|n} (1 - sign(floor(d1*d2/n))).
%F A337175 a(n) = tau^2/4 if tau is even and a(n) = (tau-1)*(tau+1)/4 if tau is odd, where tau = A000005(n) is the number of divisors of n, i.e., a(n) = A002620(A000005(n)) = floor(A000005(n)^2/4). - _Chai Wah Wu_, Jan 29 2021
%F A337175 a(n) = (A035116(n) - A010052(n))/4. - _Ridouane Oudra_, May 20 2025
%F A337175 a(n) = A337174(n) - A038548(n). - _Ridouane Oudra_, May 28 2025
%e A337175 a(9) = 2; (1,1), (1,3).
%e A337175 a(10) = 4; (1,1), (1,2), (1,5), (2,2).
%e A337175 a(11) = 1; (1,1).
%e A337175 a(12) = 9; (1,1), (1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (3,3).
%t A337175 Table[Sum[Sum[(1 - Sign[Floor[i*k/n]]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 80}]
%t A337175 a[n_] := Floor[DivisorSigma[0, n]^2/4]; Array[a, 100] (* _Amiram Eldar_, Feb 01 2025 *)
%o A337175 (Python)
%o A337175 from sympy import divisor_count
%o A337175 def A337175(n):
%o A337175     return divisor_count(n)**2//4  # _Chai Wah Wu_, Jan 29 2021
%o A337175 (PARI) a(n) = numdiv(n)^2\4; \\ _Amiram Eldar_, Feb 01 2025
%Y A337175 Cf. A000005, A002620, A038548, A337174, A035116, A010052.
%K A337175 nonn,easy
%O A337175 1,4
%A A337175 _Wesley Ivan Hurt_, Jan 28 2021