Possible new sequence based on largest prime factors of k^n+1

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Ali Sada

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Jun 5, 2026, 9:37:15 PMJun 5
to seq...@googlegroups.com

 

Hi everyone,

 

Hope all is well. I would like to propose the following sequence and I would really appreciate it if you could tell me if it’s suitable for the OEIS.

 

For a fixed positive integer n, compute the largest prime factor of k raised to the nth power plus 1 for k = 1, 2, 3, ...

The sequence term a(n) is the first value of k for which this largest prime factor is less than or equal to the previous one.

 

 

For a fixed n >= 1, let

image.png

where gpf(m) denotes the largest prime factor of m.

Starting with k = 1, compute the sequence

image.png

and continue as long as these values are strictly increasing. Let image.pngbe the first value of image.pngfor which the increase fails, i.e.,

image.png

Equivalently, a(n)-1 is the length of the initial strictly increasing run of largest prime factors of k^n+1.


The first terms are

image.png

Examples:

  • image.png:

image.png

Since image.png, the increase fails at image.png, hence image.png.

  • image.png:

image.png

coming from

image.png

Since 7 <13 , the increase fails at k=5, hence a(3) = 5.

  • image.png:

image.png

coming from

image.png

Since 1201 < 1297 the increase fails at k = 7, hence a(4) = 7.

 

First terms:

image.png

Conjecture. For every nonconstant integer polynomial whose values are eventually greater than 1, the largest prime factor of the polynomial values has infinitely many descents.

 

A similar sequence (and conjecture) could be obtained from n^k+1.

 

Best,


Ali

 

 

 


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