A337331 Number of pairs of divisors of n, (d1,d2), with d1 <= d2, whose average is an integer and divides n.

1, 2, 2, 3, 2, 5, 2, 4, 3, 4, 2, 9, 2, 4, 5, 5, 2, 8, 2, 6, 4, 4, 2, 13, 3, 4, 4, 7, 2, 12, 2, 6, 4, 4, 4, 15, 2, 4, 4, 9, 2, 10, 2, 6, 9, 4, 2, 17, 3, 6, 4, 6, 2, 11, 4, 10, 4, 4, 2, 23, 2, 4, 6, 7, 4, 11, 2, 6, 4, 8, 2, 22, 2, 4, 8, 6, 4, 10, 2, 12, 5, 4, 2, 21, 4, 4, 4, 8, 2, 22, 5, 6, 4, 4, 4, 21, 2, 6, 6, 9

Offset: 1

Author

Wesley Ivan Hurt, Aug 23 2020

Keywords

• nonn

Comments

a(n) >= d(n), where d is the number of divisors of n (A000005).

From Bernard Schott, Aug 24 2020: (Start)

a(n) = 2 if and only if n is prime.

Also, a(2^k) = k+1 for k >= 0. (End)

Note that the other divisor does not necessarily have to divide the other, for example for n=60, we also have pairs d1=3, d2=5 and d1=4, d2=6 as possible solutions - Antti Karttunen, Nov 27 2024

Examples

			a(6) = 5; The divisors of 6 are {1,2,3,6}. The pairs of divisors, (d1,d2), with d1 <= d2, whose average divides 6 are: (1,1), (1,3), (2,2), (3,3) and (6,6). So a(6) = 5.
a(7) = 2; The divisors of 7 are {1,7}. The pairs of divisors, (d1,d2), with d1 <= d2, whose average divides 7 are: (1,1) and (7,7). So a(7) = 2.
a(8) = 4; The divisors of 8 are {1,2,4,8}. The pairs of divisors, (d1,d2), with d1 <= d2, whose average divides 8 are: (1,1), (2,2), (4,4), and (8,8). So a(8) = 4.
a(9) = 3; The divisors of 9 are {1,3,9}. The pairs of divisors, (d1,d2), with d1 <= d2, whose average divides 9 are: (1,1), (3,3) and (9,9). So a(9) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005.

Programs

Mathematica

Table[Sum[Sum[(1 - Ceiling[(i + k)/2] + Floor[(i + k)/2]) (1 - Ceiling[2 n/(i + k)] + Floor[2 n/(i + k)]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]

PARI

A337331(n) = { my(divs=divisors(n), d1, d2); sum(i=1, #divs, d1=divs[i]; sum(j=i, #divs, d2=divs[j]; !((2*n)%(d1+d2)) * !((d1+d2)%2))); }; \\ Antti Karttunen, Nov 27 2024

Formula

a(n) = Sum_{d1|n, d2|n, d1<=d2, (d1+d2)|(2*n), 2|(d1+d2)} 1.

Extensions

Definition and formula clarified, and more terms added by Antti Karttunen, Nov 27 2024