Question

If $s = 4m - 3$, then $\sigma(p^s) = (1 + p^{2m-1})(1 + p + \ldots + p^{2m-2})$. Is there a similar factorization for $\sigma(q^t)$ when $t=4n$?

score 50 · Answer 1 · 2026-03-28 09:57:39

50

Views

If $s = 4m - 3$, then $\sigma(p^s) = (1 + p^{2m-1})(1 + p + \ldots + p^{2m-2})$. Is there a similar factorization for $\sigma(q^t)$ when $t=4n$?

Published on 28 Mar 2026 - 9:57

score 105 · Answer 2 · 2026-03-27 01:04:19

105

Views

How can we show this relationship between the sum of divisors function and the sum $p^{m}+2p^{m-1}+3p^{m-2}+\ldots+(m+1)$?

Published on 27 Mar 2026 - 1:04

score 66 · Answer 3 · 2026-03-26 21:27:42

66

Views

Why does this identity with the product $\prod_{p\mid n}(p+k)$ and recursive sum of divisors function is true?

Published on 26 Mar 2026 - 21:27

score 254 · Answer 4 · 2022-03-19 03:22:38

254

Views

Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?

Published on 19 Mar 2022 - 3:22

score 182 · Answer 5 · 2022-03-23 11:28:50

182

Views

Are there infinite many squarefree numbers with $\ \varphi(n)\mid \sigma(n)\ $?

Published on 23 Mar 2022 - 11:28

score 54 · Answer 6 · 2026-03-26 21:25:49

54

Views

How can we show this relationship between the recursive sum of divisors function and Figurate Number Polynomial on Primes?

Published on 26 Mar 2026 - 21:25

score 68 · Answer 7 · 2022-04-13 10:24:24

68

Views

Largest possible prime factor for given $k$?

Published on 13 Apr 2022 - 10:24

score 81 · Answer 8 · 2026-03-27 22:59:24

81

Views

Reference request regarding odd 4-perfect numbers

Published on 27 Mar 2026 - 22:59

score 95 · Answer 9 · 2026-03-27 22:57:32

95

Views

Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?

Published on 27 Mar 2026 - 22:57

score 93 · Answer 10 · 2026-02-23 01:21:44

93

Views

On Carmichael function and aliquot parts of odd perfect numbers

Published on 23 Feb 2026 - 1:21

score 102 · Answer 11 · 2026-03-27 22:53:54

102

Views

What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

Published on 27 Mar 2026 - 22:53

score 77 · Answer 12 · 2026-03-27 23:00:53

77

Views

Follow-up to MSE question 3738458

Published on 27 Mar 2026 - 23:00

score 80 · Answer 13 · 2022-05-31 17:51:47

80

Views

Finding the Ceiling function of $(\sqrt{3}+ 1)^{2n}$

Published on 31 May 2022 - 17:51

score 120 · Answer 14 · 2026-03-27 22:53:57

120

Views

Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?

Published on 27 Mar 2026 - 22:53

score 69 · Answer 15 · 2026-03-26 19:38:05

69

Views

Hand calculation of divisor summatory function

Published on 26 Mar 2026 - 19:38

List Question

If $s = 4m - 3$, then $\sigma(p^s) = (1 + p^{2m-1})(1 + p + \ldots + p^{2m-2})$. Is there a similar factorization for $\sigma(q^t)$ when $t=4n$?

How can we show this relationship between the sum of divisors function and the sum $p^{m}+2p^{m-1}+3p^{m-2}+\ldots+(m+1)$?

Why does this identity with the product $\prod_{p\mid n}(p+k)$ and recursive sum of divisors function is true?

Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?

Are there infinite many squarefree numbers with $\ \varphi(n)\mid \sigma(n)\ $?

How can we show this relationship between the recursive sum of divisors function and Figurate Number Polynomial on Primes?

Largest possible prime factor for given $k$?

Reference request regarding odd 4-perfect numbers

Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?

On Carmichael function and aliquot parts of odd perfect numbers

What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

Follow-up to MSE question 3738458

Finding the Ceiling function of $(\sqrt{3}+ 1)^{2n}$

Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?

Hand calculation of divisor summatory function

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