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# Cyclotomic Polynomials at x=n, n! and sigma(n)

There are many sequences in the OEIS dealing with cyclotomic polynomials, factorials and the divisor function, sigma.

Selected sequences and those related to prime terms are summarized here.

A000203 a(n)=sigma(n): lists many references and links.

The table below summarizes sequences in the OEIS related to prime terms of the cyclotomic polynomials,${\displaystyle \Phi _{k}(X)}$ with ${\displaystyle X=n,n!and\sigma (n)}$.

## Related Sequences

Column headings in the table below:

k                  Cyclotomic index
${\displaystyle \Phi _{k}(X)}$                Polynomial: ${\displaystyle \Phi _{k}(X)}$

${\displaystyle \Phi _{k}(n)}$                A-number of sequence listing values where ${\displaystyle x=n}$
Indices ${\displaystyle \Phi _{k}(n)}$        A-number of sequence listing indices of primes where ${\displaystyle x=n}$
Primes ${\displaystyle \Phi _{k}(n)}$         A-number of sequence listing primes where ${\displaystyle x=n}$

${\displaystyle \Phi _{k}(n!)}$               A-number of sequence listing values where ${\displaystyle x=n!}$
Indices ${\displaystyle \Phi _{k}(n!)}$       A-number of sequence listing indices of primes where ${\displaystyle x=n!}$
Primes ${\displaystyle \Phi _{k}(n!)}$        A-number of sequence listing primes where ${\displaystyle x=n!}$

${\displaystyle \Phi _{k}(\sigma (n))}$              A-number of sequence listing values where ${\displaystyle x=\sigma (n)}$
Indices ${\displaystyle \Phi _{k}(\sigma (n))}$      A-number of sequence listing indices of primes where ${\displaystyle x=\sigma (n)}$
Primes ${\displaystyle \Phi _{k}(\sigma (n))}$       A-number of sequence listing primes where ${\displaystyle x=\sigma (n)}$



Sequences
k Indices Primes Indices Primes Indices Primes
1 A023443 A008864 A000040 A033312 A002982 A055490 A039653 A248792 A248793
2 A000027 A006093 A000040 A038507 A002981 A088332 A088580 A065512 A258430
3 A002061(n+1) A002384 A002383 A066142 A051856 A162455 A258774 A258775 A258776
4 A002522 A005574 A002496 A020549 A046029 A051739 A258974 A258976 A258977
5 A053699 A049409 A088548 A259264 A200906 A259265 A258978 A258979 A258980
6 A002061 A055494 A002383 A051857 A259184 A259185 A259186
7 A053716 A100330 A088550 A258357 A259251 A259252 A259253
8 A123865 A000068 A037896 A259308 A259309 A259310
9 A060883 A153439 A162601 A258355 A259369 A259370 A259371
10 A060884 A246392 A259257 A258356 A259410 A259411 A259412