cp's OEIS Frontend

A337174 Number of pairs of divisors of n (d1,d2) such that d1 <= d2 and d1*d2 >= n.

%I A337174 #35 Jun 06 2025 11:56:08
%S A337174 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 A337174 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 A337174 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 A337174 Number of pairs of divisors of n (d1,d2) such that d1 <= d2 and d1*d2 >= n.
%H A337174 Amiram Eldar, <a href="/A337174/b337174.txt">Table of n, a(n) for n = 1..10000</a>
%F A337174 a(n) = Sum_{d1|n, d2|n} sign(floor(d1*d2/n)).
%F A337174 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
%F A337174 a(n) = A337175(n) + A038548(n). - _Ridouane Oudra_, May 24 2025
%F A337174 a(n) = (2*d^2 + 4*d + 1 - (-1)^d)/8, where d = A000005(n). (see Wu formula)
%e A337174 a(4) = 4; (1,4), (2,2), (2,4), (4,4).
%e A337174 a(5) = 2; (1,5), (5,5).
%e A337174 a(6) = 6; (1,6), (2,3), (2,6), (3,3), (3,6), (6,6).
%t A337174 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}]
%t A337174 a[n_] := Floor[(DivisorSigma[0, n]+1)^2/4]; Array[a, 100] (* _Amiram Eldar_, Feb 02 2025 *)
%o A337174 (Python)
%o A337174 from sympy import divisor_count
%o A337174 def A337174(n):
%o A337174     return (divisor_count(n)+1)**2//4 # _Chai Wah Wu_, Jan 29 2021
%o A337174 (PARI) a(n) = (numdiv(n)+1)^2\4; \\ _Amiram Eldar_, Feb 02 2025
%Y A337174 Cf. A000005, A002620, A337175, A038548.
%K A337174 nonn,easy
%O A337174 1,2
%A A337174 _Wesley Ivan Hurt_, Jan 28 2021