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Odd divisor function

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The odd divisor function for a positive integer is defined as the sum of the th powers of the odd divisors of

The number of odd divisors of n is represented by sequence A001227 and the sum of odd divisors of n is represented by sequence A000593.

Formulae for the odd divisor function

The odd divisor function is multiplicative with

for odd primes .[1]

Generating function of the odd divisor function

The o.g.f. for the odd divisor function is

Dirichlet generating function of the odd divisor function

The Dirichlet g.f. for the odd divisor function is

.[2]

There are three ways to regroup the three factors on the right hand side in pairs, and each of these factorizations implies that the odd divisor function is a Dirichlet convolution of two other integer sequences. One example:

  • : Dirichlet g.f. . Dirichlet convolution of A176415 and A000578.

Number of odd divisors function

The number of odd divisors function for a positive integer is defined as the count of the odd divisors of

Number of odd divisors greater than 1, number of odd divisors smaller than n

(...)

Formulae for the number of odd divisors function

Generating function for number of odd divisors function

The o.g.f. for the number of odd divisors function is

Dirichlet generating function for number of odd divisors function

The formula is obtained by inserting into the formula mentioned above

.[2]

Number of ways of factoring n with all factors odd and greater than 1

(...)

Sum of odd divisors function

The sum of odd divisors function for a positive integer is defined as the sum of the odd divisors of

Formulae for the sum of odd divisors function

(...)

Generating function for sum of odd divisors function

(...)

Logarithmic generating function for sum of odd divisors function

A possible new L.g.f. for the sum of odd divisors function (proposed by Peter Lawrence in the SeqFan mailing list of June 2011[3]) is

Dirichlet generating function for sum of odd divisors function

The formula is obtained inserting into the generic formula shown above.

Sum of aliquot odd divisors of n

"Untouchable numbers"? (sum of odd divisors function)

(...)

Sum of odd divisors of n equal to n+1

(...)

The Lawrence conjecture

An observation of Peter Lawrence in the SeqFan mailing list of June 2011[4] is that

Use the multiplicativity mentioned above to split off the powers of 2 leads to a variant of the question whether quasiperfect numbers exist.

"Perfect numbers"? (sum of odd divisors function)

? An odd divisors equivalent of the "perfect numbers" concept could be defined requiring

There are certainly no solutions with powers of two to this equation, because in these cases. But what about even numbers? For odd , the equation is the same as as required for odd perfect numbers, unlikely to exist (see A000396).

"k-perfect numbers"? (sum of odd divisors function)

? An odd divisors equivalent of the "k-perfect numbers" concept could be defined requiring

"Deficient numbers"? (sum of odd divisors function)

? An odd divisors equivalent of the "deficient numbers" concept could be defined requiring

"Abundant numbers"? (sum of odd divisors function)

? An odd divisors equivalent of the "abundant numbers" concept could be defined requiring

The satisfying this criterion are exactly the (ordinary) abundant numbers A005231.

Table of related formulae and values

(...)

Table of sequences

Odd divisor function sequences
sequences A-number
0 {1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, ...} A001227
1 {1, 1, 4, 1, 6, 4, 8, 1, 13, 6, 12, 4, 14, 8, 24, 1, 18, 13, 20, 6, 32, 12, 24, 4, 31, 14, 40, 8, 30, 24, 32, 1, 48, 18, 48, 13, 38, 20, 56, 6, 42, 32, 44, 12, 78, 24, 48, 4, ...} A000593
2 {1, 1, 10, 1, 26, 10, 50, 1, 91, 26, 122, 10, 170, 50, 260, 1, 290, 91, 362, 26, 500, 122, 530, 10, 651, 170, 820, 50, 842, 260, 962, 1, 1220, 290, 1300, 91, 1370, ...} A050999
3 {1, 1, 28, 1, 126, 28, 344, 1, 757, 126, 1332, 28, 2198, 344, 3528, 1, 4914, 757, 6860, 126, 9632, 1332, 12168, 28, 15751, 2198, 20440, 344, 24390, 3528, 29792, ...} A051000
4 {1, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, 14642, 82, 28562, 2402, 51332, 1, 83522, 6643, 130322, 626, 196964, 14642, 279842, 82, 391251, 28562, 538084, 2402, ...} A051001
5 {1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, 161052, 244, 371294, 16808, 762744, 1, 1419858, 59293, 2476100, 3126, 4101152, 161052, 6436344, 244, ...} A051002
6 {1, 1, 730, 1, 15626, ...} A??????
7 {1, 1, 2188, 1, 78126, ...} A??????
8 {1, 1, 6562, 1, 390626, , ...} A??????
9 {1, 1, 19684, 1, 1953126, ...} A??????
10 {1, 1, 59050, 1, 9765626, ...} A??????
11 {1, 1, 177148, 1, 48828126, ...} A??????
12 {1, 1, 531442, 1, 244140626, ...} A??????


Sequences

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See also

Notes

  1. R. J. Mathar, arXiv:1106.4038.
  2. 2.0 2.1 R. J. Mathar, arXiv:1106.4038, chapter 3.12.
  3. Peter Lawrence, Possible new G.F. for A000593 "sum of odd divisors of N", June 18 2011.
  4. Peter Lawrence, Number theory question about A000593, Jun 18 2011.