A333707 Number of ways to write n as the sum of two distinct positive integers that have the same number of divisors.

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 3, 0, 3, 1, 3, 2, 3, 1, 4, 2, 2, 1, 3, 3, 5, 2, 5, 2, 5, 2, 7, 2, 2, 2, 6, 4, 6, 4, 6, 3, 5, 3, 11, 5, 6, 0, 6, 4, 9, 3, 7, 3, 4, 3, 12, 7, 5, 4, 10, 6, 10, 3, 7, 3, 10, 3, 16, 8, 6, 4, 8, 6, 11, 5, 11, 4, 8, 5

Offset

1,13

Links

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

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} [d(i) = d(n-i)], where [] is the Iverson bracket and d is the number of divisors of n (A000005).

Example

a(16) = 3; There are 3 ways to write 16 as the sum of 2 distinct numbers with the same number of divisors: 16 = 13+3 (13 and 3 both have 2 divisors), 16 = 11+5 (11 and 5 both have 2 divisors), 16 = 10+6 (10 and 6 both have 4 divisors).

Mathematica

Table[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, n - i]], {i, Floor[(n - 1)/2]}], {n, 100}]

Crossrefs

Cf. A000005, A333626 (not necessarily distinct).

Sequence in context: A117046 A268192 A362312 * A077653 A077889 A305805

Adjacent sequences: A333704 A333705 A333706 * A333708 A333709 A333710

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 02 2020

Status

approved