Question

Do the numbers preceding primes have on an average fewer divisors than the numbers succeeding primes?

score 144 · Answer 1 · 2020-06-16 20:42:13

144

Views

Do the numbers preceding primes have on an average fewer divisors than the numbers succeeding primes?

Published on 16 Jun 2020 - 20:42

score 40 · Answer 2 · 2020-06-17 04:49:00

40

Views

Polynomial properties

Published on 17 Jun 2020 - 4:49

score 173 · Answer 3 · 2020-07-02 14:30:44

173

Views

Prove that $ a_{1}^{3}+a_{2}^{3}+\cdots+a_{l}^{3}=\left(a_{1}+a_{2}+\cdots+a_{l}\right)^{2} $

Published on 02 Jul 2020 - 14:30

score 96 · Answer 4 · 2026-03-26 04:34:57

96

Views

Does $k=1$ follow from $I(5^k)+I(m^2) \leq \frac{43}{15}$, if $p^k m^2$ is an odd perfect number with special prime $p=5$?

Published on 26 Mar 2026 - 4:34

score 129 · Answer 5 · 2026-03-28 03:09:54

129

Views

On the abundancy index of divisors of odd perfect numbers and a possible upper bound for the special/Euler prime

Published on 28 Mar 2026 - 3:09

score 554 · Answer 6 · 2020-07-16 10:38:04

554

Views

There is only one positive integer that is both the product and sum of all its proper positive divisors, and that number is $6$.

Published on 16 Jul 2020 - 10:38

score 59 · Answer 7 · 2026-03-26 06:01:39

59

Views

On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$

Published on 26 Mar 2026 - 6:01

score 46 · Answer 8 · 2020-07-24 03:41:38

46

Views

For which primes $p$ and positive integers $k$ is the deficiency $D(p^k)$ equal to the arithmetic derivative of $p^k$?

Published on 24 Jul 2020 - 3:41

score 72 · Answer 9 · 2026-03-26 07:59:21

72

Views

If $a$ is deficient-perfect, can $ab$ be deficient-perfect if $b > 1$, and $ab$ is odd?

Published on 26 Mar 2026 - 7:59

score 275 · Answer 10 · 2016-07-05 23:07:47

275

Views

What is the complete (polynomial) factorization of $\sigma(p^k)$, where $p$ is prime with $p \equiv k \equiv 1 \pmod 4$?

Published on 05 Jul 2016 - 23:07

score 100 · Answer 11 · 2016-07-09 19:46:20

100

Views

Is there an interpretation for why $\sigma(x) = x \iff x = 1$, that is not (purely) number-theoretic?

Published on 09 Jul 2016 - 19:46

score 132 · Answer 12 · 2026-03-26 05:54:22

132

Views

Can an odd perfect number be divisible by either $2049$ or $2051$?

Published on 26 Mar 2026 - 5:54

score 97 · Answer 13 · 2026-03-26 06:01:40

97

Views

If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Published on 26 Mar 2026 - 6:01

score 41 · Answer 14 · 2016-07-15 17:25:07

41

Views

How to find that a number is a sum of multiple of different numbers?

Published on 15 Jul 2016 - 17:25

score 79 · Answer 15 · 2026-03-26 07:33:33

79

Views

Combining a working hypothesis for odd perfect numbers with an inequality for logarithms

Published on 26 Mar 2026 - 7:33

List Question

Do the numbers preceding primes have on an average fewer divisors than the numbers succeeding primes?

Polynomial properties

Prove that $ a_{1}^{3}+a_{2}^{3}+\cdots+a_{l}^{3}=\left(a_{1}+a_{2}+\cdots+a_{l}\right)^{2} $

Does $k=1$ follow from $I(5^k)+I(m^2) \leq \frac{43}{15}$, if $p^k m^2$ is an odd perfect number with special prime $p=5$?

On the abundancy index of divisors of odd perfect numbers and a possible upper bound for the special/Euler prime

There is only one positive integer that is both the product and sum of all its proper positive divisors, and that number is $6$.

On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$

For which primes $p$ and positive integers $k$ is the deficiency $D(p^k)$ equal to the arithmetic derivative of $p^k$?

If $a$ is deficient-perfect, can $ab$ be deficient-perfect if $b > 1$, and $ab$ is odd?

What is the complete (polynomial) factorization of $\sigma(p^k)$, where $p$ is prime with $p \equiv k \equiv 1 \pmod 4$?

Is there an interpretation for why $\sigma(x) = x \iff x = 1$, that is not (purely) number-theoretic?

Can an odd perfect number be divisible by either $2049$ or $2051$?

If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

How to find that a number is a sum of multiple of different numbers?

Combining a working hypothesis for odd perfect numbers with an inequality for logarithms

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