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A299480
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List of pairs (a,b) where in the n-th pair, a = number of odd divisors of n and b = number of even divisors of n.
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10
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1, 0, 1, 1, 2, 0, 1, 2, 2, 0, 2, 2, 2, 0, 1, 3, 3, 0, 2, 2, 2, 0, 2, 4, 2, 0, 2, 2, 4, 0, 1, 4, 2, 0, 3, 3, 2, 0, 2, 4, 4, 0, 2, 2, 2, 0, 2, 6, 3, 0, 2, 2, 4, 0, 2, 4, 2, 0, 4, 4, 2, 0, 1, 5, 4, 0, 2, 2, 4, 0, 3, 6, 2, 0, 2, 2, 4, 0, 2, 6, 2, 0, 4, 4, 2, 0, 2, 4, 6, 0, 2, 2, 2, 0, 2, 8, 3, 0, 3, 3, 4, 0, 2, 4
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OFFSET
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1,5
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COMMENTS
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Also sequence found by reading in the upper 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 upper part of the diagram equals A001227(n) the number of odd divisors of n. The number of curves that emerge from the same point (n, 0) to the right hand in the upper part of the diagram equals A183063(n) the number of even divisors of n. So the n-th pair is (A001227(n), A183063(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 upper part from the first diagram.
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LINKS
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FORMULA
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G.f.: Sum_{n>=1} (x^(2*n-1) + x^(4*n))/(1-x^(4*n)). - Robert Israel, Feb 11 2018
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EXAMPLE
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Array begins:
1 1 0
2 1 1
3 2 0
4 1 2
5 2 0
6 2 2
7 2 0
8 1 3
9 3 0
10 2 2
11 2 0
12 2 4
...
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MAPLE
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f := proc (n) local t; t := numtheory:-tau(n/2^padic:-ordp(n, 2)); t, numtheory:-tau(n)-t end proc:
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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