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

enzocreti · 2020-09-29, 09:26

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?

LaurV · 2020-09-29, 10:24

wow! you advanced now! this is a step up from the pg(whatever) numbers

axn · 2020-09-29, 11:01

Quote:

Originally Posted by LaurV

wow! you advanced now! this is a step up from the pg(whatever) numbers

69660 is a pg prime thingie. Sorry to disappoint you.

Dr Sardonicus · 2020-09-29, 12:12

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.

2020-09-29, 09:26	#1
enzocreti Mar 2018 7²·11 Posts	*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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2020-09-29, 10:24	#2
LaurV Romulan Interpreter "name field" Jun 2011 Thailand 2822₁₆ Posts	wow! you advanced now! this is a step up from the pg(whatever) numbers

2020-09-29, 12:12	#4
Dr Sardonicus Feb 2017 Nowhere 3·19·109 Posts	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.