Odd divisor function

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The odd divisor function $\scriptstyle \sigma _{k}^{(o)}(n)\,$ for a positive integer $\scriptstyle n\,$ is defined as the sum of the $\scriptstyle k\,$^th powers of the odd divisors of $\scriptstyle n\,$

$\sigma _{k}^{(o)}(n)\equiv \sum _{2\nmid d,\,d|n}d^{k},\quad k\geq 0.\,$

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

$\sigma _{k}^{(o)}(2^{e})=1,\quad k\geq 0,\,e\geq 0;\,$

$\sigma _{k}^{(o)}(p^{e})=\sigma _{k}(p^{e})={\begin{cases}e+1,&k=0,e\geq 0,\\{\frac {p^{k(e+1)}-1}{p^{k}-1}},&k>0,e\geq 0,\\\end{cases}}$

for odd primes $\scriptstyle p\,$.

Generating function of the odd divisor function

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

$G_{\{\sigma _{k}^{(o)}(n)\}}(x)\equiv \sum _{n=1}^{\infty }\sigma _{k}^{(o)}(n)~x^{n}=~?.\,$

Dirichlet generating function of the odd divisor function

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

$D_{\{\sigma _{k}^{(o)}(n)\}}(s)\equiv \sum _{n=1}^{\infty }{\frac {\sigma _{k}^{(o)}(n)}{n^{s}}}=(1-2^{k-s})\zeta (s)\zeta (s-k),\quad k\geq 0.\,$.

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:

• $\scriptstyle k\,=\,3\,$: Dirichlet g.f. $\scriptstyle (1-2^{(3-s)})\zeta (s)\zeta (s-3)\,$. Dirichlet convolution of $\scriptstyle (-1)^{n}\,$ A176415$\scriptstyle (n)\,$ and A000578.

Number of odd divisors function

The number of odd divisors function $\scriptstyle \tau ^{(o)}(n)\,\equiv \,\sigma _{0}^{(o)}(n)\,$ for a positive integer $\scriptstyle n\,$ is defined as the count of the odd divisors of $\scriptstyle n\,$

$\tau ^{(o)}(n)\equiv \sigma _{0}^{(o)}(n)=\sum _{2\nmid d,\,d|n}1,\quad n\geq 1.\,$

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

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Formulae for the number of odd divisors function

$\sigma _{0}^{(o)}(n)={\begin{cases}\sigma _{0}(n)&{\text{if }}n{\text{ is odd}},\\\sigma _{0}(n)-\sigma _{0}({\frac {n}{2}})&{\text{if }}n{\text{ is even}}.\end{cases}}$

Generating function for number of odd divisors function

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

$G_{\{\sigma _{0}^{(o)}(n)\}}(x)\equiv \sum _{n=1}^{\infty }\sigma _{0}^{(o)}(n)~x^{n}=\sum _{n=1}^{\infty }{\frac {x^{n}}{1-x^{2n}}}.\,$

Dirichlet generating function for number of odd divisors function

The formula is obtained by inserting $\scriptstyle k\,=\,0\,$ into the formula mentioned above

$D_{\{\sigma _{0}^{(o)}(n)\}}(s)\equiv \sum _{n=1}^{\infty }{\frac {\sigma _{0}^{(o)}(n)}{n^{s}}}=(1-2^{-s})(\zeta (s))^{2},\quad k\geq 0.\,$.

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

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Sum of odd divisors function

The sum of odd divisors function $\scriptstyle \sigma ^{(o)}(n)\,\equiv \,\sigma _{1}^{(o)}(n)\,$ for a positive integer $\scriptstyle n\,$ is defined as the sum of the odd divisors of $\scriptstyle n\,$

$\sigma ^{(o)}(n)\equiv \sigma _{1}^{(o)}(n)=\sum _{2\nmid d,\,d|n}d,\quad n\geq 1.\,$

Formulae for the sum of odd divisors function

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Generating function for sum of odd divisors function

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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) is

$L_{\{\sigma _{k}^{(o)}(n)\}}(x)\equiv \sum _{n=1}^{\infty }\sigma _{k}^{(o)}(n){\frac {x^{n}}{n}}=\sum _{n=1}^{\infty }\log {\bigg (}1+{\frac {1}{x^{n}}}{\bigg )}\,$

Dirichlet generating function for sum of odd divisors function

The formula is obtained inserting $\scriptstyle k\,=\,1\,$ into the generic formula shown above.

Sum of aliquot odd divisors of n

$s^{(o)}(n)\equiv {\begin{cases}\sigma ^{(o)}(n)&,n{\text{ even, }}\\\sigma ^{(o)}(n)-n&,n{\text{ odd. }}\end{cases}}$

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

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Sum of odd divisors of n equal to n+1

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The Lawrence conjecture

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

$\sigma _{1}^{(o)}(n)=n+1{\overset {?}{\iff }}n{\rm {~is~an~odd~prime}}.\,$

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

$\sigma ^{(o)}(n)=2n,\quad n\geq 1.\,$

There are certainly no solutions with powers of two $\scriptstyle 2^{n}\,$ to this equation, because $\scriptstyle \sigma ^{(o)}(2^{n})\,=\,1\,$ in these cases. But what about even numbers? For odd $\scriptstyle n\,$, 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

$\sigma ^{(o)}(n)=kn,\quad k\geq 2,\,n\geq 1.\,$

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

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

$\sigma ^{(o)}(n)<2n,\quad n\geq 1.\,$

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

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

$\sigma ^{(o)}(n)>2n,\quad n\geq 1.\,$

The $\scriptstyle n\,$ satisfying this criterion are exactly the (ordinary) abundant numbers A005231.

Table of related formulae and values

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Table of sequences

Odd divisor function sequences

$k\,$	$\sigma _{k}^{(o)}(n),\ n\geq 1,\,$ 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$\scriptstyle (n)\,$
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$\scriptstyle (n)\,$
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$\scriptstyle (n)\,$
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$\scriptstyle (n)\,$
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$\scriptstyle (n)\,$
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$\scriptstyle (n)\,$
6	{1, 1, 730, 1, 15626, ...}	A??????$\scriptstyle (n)\,$
7	{1, 1, 2188, 1, 78126, ...}	A??????$\scriptstyle (n)\,$
8	{1, 1, 6562, 1, 390626, , ...}	A??????$\scriptstyle (n)\,$
9	{1, 1, 19684, 1, 1953126, ...}	A??????$\scriptstyle (n)\,$
10	{1, 1, 59050, 1, 9765626, ...}	A??????$\scriptstyle (n)\,$
11	{1, 1, 177148, 1, 48828126, ...}	A??????$\scriptstyle (n)\,$
12	{1, 1, 531442, 1, 244140626, ...}	A??????$\scriptstyle (n)\,$

Sequences

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See also

• Divisor function
• Unitary variants in A068068 and A192066