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

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, 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, 12, 4, 16, 1, 9, 4, 16, 1, 36, 1, 4, 9, 9, 4, 16, 1, 25, 6, 4, 1, 36, 4, 4

Offset: 1

Author

Wesley Ivan Hurt, Jan 28 2021

Keywords

• nonn
• easy

Examples

			a(9) = 2; (1,1), (1,3).
a(10) = 4; (1,1), (1,2), (1,5), (2,2).
a(11) = 1; (1,1).
a(12) = 9; (1,1), (1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (3,3).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A002620, A038548, A337174, A035116, A010052.

Programs

Mathematica

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}]
a[n_] := Floor[DivisorSigma[0, n]^2/4]; Array[a, 100] (* Amiram Eldar, Feb 01 2025 *)

PARI

a(n) = numdiv(n)^2\4; \\ Amiram Eldar, Feb 01 2025

Python

from sympy import divisor_count
def A337175(n):
    return divisor_count(n)**2//4  # Chai Wah Wu, Jan 29 2021

Formula

a(n) = Sum_{d1|n, d2|n} (1 - sign(floor(d1*d2/n))).

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

a(n) = (A035116(n) - A010052(n))/4. - Ridouane Oudra, May 20 2025

a(n) = A337174(n) - A038548(n). - Ridouane Oudra, May 28 2025