Odd Divisor Function
المؤلف:
Dickson, L. E.
المصدر:
History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover
الجزء والصفحة:
...
14-8-2020
921
Odd Divisor Function
The odd divisor function
 |
(1)
|
is the sum of 
th powers of the odd divisors of a number 
. It is the analog of the divisor function for odd divisors only.
For the case 
,
where 
is defined to be 0 if 
is odd. The generating function is given by
where 
is a Jacobi elliptic function.
Rather surprisingly, 
gives the number of factors of the polynomial 
.
The following table gives the first few 
.
 |
OEIS |
 |
| 0 |
A001227 |
1, 1, 2, 1, 2, 2, 2, 1, 3, 2, ... |
| 1 |
A000593 |
1, 1, 4, 1, 6, 4, 8, 1, 13, 6, ... |
| 2 |
A050999 |
1, 1, 10, 1, 26, 10, 50, 1, 91, 26, ... |
| 3 |
A051000 |
1, 1, 28, 1, 126, 28, 344, 1, 757, 126, ... |
| 4 |
A051001 |
1, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, ... |
| 5 |
A051002 |
1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, ... |
This function arises in Ramanujan's Eisenstein series 
and in a recurrence relation for the partition function P.
REFERENCES:
Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 306, 2005.
Hirzebruch, F. Manifolds and Modular Forms, 2nd ed. Braunschweig, Germany: Vieweg, p. 133, 1994.
Riordan, J. Combinatorial Identities. New York: Wiley, p. 187, 1979.
Sloane, N. J. A. Sequences A000593/M3197, A001227, A050999, A051000, A051001, and A051002 in "The On-Line Encyclopedia of Integer Sequences."
Verhoeff, T. "Rectangular and Trapezoidal Arrangements." J. Integer Sequences 2, #99.1.6, 1999.a
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