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Question on generalized identity for the number of divisors function

score 46 · Answer 1 · 2026-03-25 12:53:01

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Question on generalized identity for the number of divisors function

Published on 25 Mar 2026 - 12:53

score 97 · Answer 2 · 2019-11-16 05:15:35

97

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Prove that $\sum_{u = 1}^{N} \sum_{v=1}^{N} \left(\left[\sqrt{u^2-4v} \in Z\right]+\left[\sqrt{u^2+4v} \in Z\right]\right) = D \left(N\right)-1$

Published on 16 Nov 2019 - 5:15

score 70 · Answer 3 · 2019-11-24 10:45:50

70

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A possible disproof for the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Published on 24 Nov 2019 - 10:45

score 42 · Answer 4 · 2019-12-02 12:39:02

42

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Is the aliquot sum of an odd number always odd?

Published on 02 Dec 2019 - 12:39

score 73 · Answer 5 · 2019-12-03 10:22:49

73

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proving that there are no odd perfect numbers

Published on 03 Dec 2019 - 10:22

score 190 · Answer 6 · 2019-12-03 17:48:17

190

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Is there a recurrence relation that describes the aliquot sum?

Published on 03 Dec 2019 - 17:48

score 58 · Answer 7 · 2019-12-15 11:55:10

58

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On the biconditional $I(n^2) = 2 - \frac{5}{3q} \iff (k = 1 \land q = 5)$, where $q^k n^2$ is an odd perfect number

Published on 15 Dec 2019 - 11:55

score 53 · Answer 8 · 2019-12-20 12:06:55

53

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PROOF: A Relationship Between A Natural Number and The Quantity of Its Divisors' Divisors

Published on 20 Dec 2019 - 12:06

score 71 · Answer 9 · 2019-12-23 05:15:44

71

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Implications of $q \neq 1049$ when $q^k n^2$ is an odd perfect number

Published on 23 Dec 2019 - 5:15

score 76 · Answer 10 · 2020-01-14 22:43:21

76

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Develop asymptotic as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{k} \sum_{d \mid i (2k-i), d > N} 1$

Published on 14 Jan 2020 - 22:43

score 150 · Answer 11 · 2020-01-20 05:06:12

150

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On the Diophantine equation $m^2 - p^k = 4z$, where $z \in \mathbb{N}$ and $p$ is a prime satisfying $p \equiv k \equiv 1 \pmod 4$

Published on 20 Jan 2020 - 5:06

score 151 · Answer 12 · 2026-03-25 13:58:10

151

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Find the asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \left\{{\sqrt{{N}^{2}+{k}^{2}}}\right\}$

Published on 25 Mar 2026 - 13:58

score 106 · Answer 13 · 2020-01-21 23:17:07

106

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Asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} k \sum_{e \mid 2k}\frac{\Lambda \left({e}\right)}{e}$

Published on 21 Jan 2020 - 23:17

score 50 · Answer 14 · 2020-02-06 16:13:19

50

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$(d_{n + 1} - d_n)$ is a geometric progression where $d_1, d_2, \cdots, d_{\tau(m) - 1}, d_{\tau(m)} \mid m$. Prove that $m$ is a prime power.

Published on 06 Feb 2020 - 16:13

score 95 · Answer 15 · 2020-02-09 15:06:16

95

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If $q^k n^2$ is an odd perfect number with special prime $q$, then its index at the prime $q$ is not a square.

Published on 09 Feb 2020 - 15:06

List Question

Question on generalized identity for the number of divisors function

Prove that $\sum_{u = 1}^{N} \sum_{v=1}^{N} \left(\left[\sqrt{u^2-4v} \in Z\right]+\left[\sqrt{u^2+4v} \in Z\right]\right) = D \left(N\right)-1$

A possible disproof for the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Is the aliquot sum of an odd number always odd?

proving that there are no odd perfect numbers

Is there a recurrence relation that describes the aliquot sum?

On the biconditional $I(n^2) = 2 - \frac{5}{3q} \iff (k = 1 \land q = 5)$, where $q^k n^2$ is an odd perfect number

PROOF: A Relationship Between A Natural Number and The Quantity of Its Divisors' Divisors

Implications of $q \neq 1049$ when $q^k n^2$ is an odd perfect number

Develop asymptotic as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{k} \sum_{d \mid i (2k-i), d > N} 1$

On the Diophantine equation $m^2 - p^k = 4z$, where $z \in \mathbb{N}$ and $p$ is a prime satisfying $p \equiv k \equiv 1 \pmod 4$

Find the asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \left\{{\sqrt{{N}^{2}+{k}^{2}}}\right\}$

Asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} k \sum_{e \mid 2k}\frac{\Lambda \left({e}\right)}{e}$

$(d_{n + 1} - d_n)$ is a geometric progression where $d_1, d_2, \cdots, d_{\tau(m) - 1}, d_{\tau(m)} \mid m$. Prove that $m$ is a prime power.

If $q^k n^2$ is an odd perfect number with special prime $q$, then its index at the prime $q$ is not a square.

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