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

1,2

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

Links

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

Formula

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

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A000005.

Sequence in context: A033099 A330833 A378436 * A018892 A343654 A100565

Adjacent sequences: A337328 A337329 A337330 * A337332 A337333 A337334

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 23 2020

Extensions

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

Status

approved