A337174 - OEIS

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A337174

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

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

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..83.

FORMULA

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

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

EXAMPLE

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

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

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

MATHEMATICA

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}]

PROG

(Python)

from sympy import divisor_count

def A337174(n):

return (divisor_count(n)+1)**2//4 # Chai Wah Wu, Jan 29 2021

CROSSREFS