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Old 2020-09-29, 09:26   #1
Mar 2018

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Default sigma(n)=x*(x+1)^3

let be sigma(n) the sum of the divisors of n

something is known about numbers n such that

sigma(n)=x*(x+1)^3 with x positive integer?

for example n=69660, sigma(69660)=21*(22^3)

are there infinitely many examples?
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Old 2020-09-29, 10:24   #2
Romulan Interpreter
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Jun 2011

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wow! you advanced now! this is a step up from the pg(whatever) numbers
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Old 2020-09-29, 11:01   #3
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Jun 2003

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Originally Posted by LaurV View Post
wow! you advanced now! this is a step up from the pg(whatever) numbers
69660 is a pg prime thingie. Sorry to disappoint you.
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Old 2020-09-29, 12:12   #4
Dr Sardonicus
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Feb 2017

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If x is a positive integer and f(x) = x*(x+1)^3 - 1 is a prime p, then sigma(p) = p+1 = x*(x+1)^3.

Everyone knows there are infinitely many x for which f(x) is prime (Bunyakovsky and Bateman-Horn conjectures), but nobody knows how to prove it.

The smallest such x is 1.
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