Difference between revisions of "Sum of divisors functions written in terms of partition function"
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(Created page with "==Theorem== The following formula holds: $$\sigma_1(n)=p(n)+\displaystyle\sum_{1 \leq \frac{3k^2 \pm k}{2} \leq n} (-1)^k\dfrac{3k^2 \pm k}{2} p \left(n - \dfrac{3k^2 \pm k}{2...") |
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Recurrence relation for partition function with sum of divisors|next= | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Recurrence relation for partition function with sum of divisors|next=Asymptotic formula for partition function}}: $24.2.1 \mathrm{II}.B.$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 20:47, 26 June 2016
Theorem
The following formula holds: $$\sigma_1(n)=p(n)+\displaystyle\sum_{1 \leq \frac{3k^2 \pm k}{2} \leq n} (-1)^k\dfrac{3k^2 \pm k}{2} p \left(n - \dfrac{3k^2 \pm k}{2} \right),$$ where $\sigma_1$ denotes the sum of divisors function and $p$ denotes the partition function.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.2.1 \mathrm{II}.B.$
