A337174 - OEIS

%I #24 Jan 30 2021 23:24:38

%S 1,2,2,4,2,6,2,6,4,6,2,12,2,6,6,9,2,12,2,12,6,6,2,20,4,6,6,12,2,20,2,

%T 12,6,6,6,25,2,6,6,20,2,20,2,12,12,6,2,30,4,12,6,12,2,20,6,20,6,6,2,

%U 42,2,6,12,16,6,20,2,12,6,20,2,42,2,6,12,12,6,20,2,30,9,6,2

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

%F a(n) = Sum_{d1|n, d2|n} sign(floor(d1*d2/n)).

%F a(n) = tau*(tau+2)/4 if tau is even and a(n) = (tau+1)^2/4 if tau is odd, where tau = A000005(n) is the number of divisors of n, i.e., a(n) = A002620(A000005(n)+1) = floor((A000005(n)+1)^2/4). - _Chai Wah Wu_, Jan 29 2021

%e a(4) = 4; (1,4), (2,2), (2,4), (4,4).

%e a(5) = 2; (1,5), (5,5).

%e a(6) = 6; (1,6), (2,3), (2,6), (3,3), (3,6), (6,6).

%t Table[Sum[Sum[Sign[Floor[i*k/n]] (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]

%o (Python)

%o from sympy import divisor_count

%o def A337174(n):

%o     return (divisor_count(n)+1)**2//4 # _Chai Wah Wu_, Jan 29 2021

%Y Cf. A000005, A002620, A337175.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Jan 28 2021