A299485 List of pairs (a,b) where in the n-th pair, a = number of even divisors of n and b = number of odd divisors of n.

0, 1, 1, 1, 0, 2, 2, 1, 0, 2, 2, 2, 0, 2, 3, 1, 0, 3, 2, 2, 0, 2, 4, 2, 0, 2, 2, 2, 0, 4, 4, 1, 0, 2, 3, 3, 0, 2, 4, 2, 0, 4, 2, 2, 0, 2, 6, 2, 0, 3, 2, 2, 0, 4, 4, 2, 0, 2, 4, 4, 0, 2, 5, 1, 0, 4, 2, 2, 0, 4, 6, 3, 0, 2, 2, 2, 0, 4, 6, 2, 0, 2, 4, 4, 0, 2, 4, 2, 0, 6, 2, 2, 0, 2, 8, 2, 0, 3, 3, 3, 0, 4, 4, 2

Offset

1,6

Comments

Also sequence found by reading in the lower part of the diagram of periodic curves for the number of divisors of n (see the first diagram in the Links section). Explanation: the number of curves that emerge from the point (n, 0) to the left hand in the lower part of the diagram equals A183063(n) the number of even divisors of n. The number of curves that emerge from the same point (n, 0) to the right hand in the lower part of the diagram equals A001227(n) the number of odd divisors of n. So the n-th pair is (A183063(n), A001227(n)). Also the total number of curves that emerges from the same point (n, 0) equals A000005(n), the number of divisors of n. Note that at the point (n, 0) the inflection point of the curve that emerges with diameter k represents the divisor n/k.

The second diagram in the links section shows only the lower part from the first diagram, upside down.

Links

Table of n, a(n) for n=1..104.

Omar E. Pol, Diagram of periodic curves for the number of divisors of n (for n = 1..16)

Omar E. Pol, Lower part upside down of the diagram of periodic curves for the number of divisors of n (for n = 1..16)

Formula

Pair(a,b) = Pair(A183063(n), A001227(n)).

Example

Array begins:
n      A183063  A001227
1         0        1
2         1        1
3         0        2
4         2        1
5         0        2
6         2        2
7         0        2
8         3        1
9         0        3
10        2        2
11        0        2
12        4        2
...

Mathematica

Array[{#2, #1 - #2} & @@ {DivisorSigma[0, #], DivisorSum[#, 1 &, EvenQ]} &, 52] // Flatten (* Michael De Vlieger, Mar 04 2018 *)

Crossrefs

Another version of A299480.

Row sums give A000005.

Cf. A001227, A027750, A048272, A183063.

Sequence in context: A287200 A284387 A143667 * A246785 A084934 A125927

Adjacent sequences: A299482 A299483 A299484 * A299486 A299487 A299488

Keyword

nonn,tabf

Author

Omar E. Pol, Mar 03 2018

Status

approved