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

Author

Omar E. Pol, Mar 03 2018

Keywords

• nonn
• tabf

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.

Examples

			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
...
		

Links

• 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)

Crossrefs

Another version of A299480.

Row sums give A000005.

Cf. A001227, A027750, A048272, A183063.

Programs

Mathematica

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

Formula

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