A070519 -id:A070519 - cp's OEIS frontend

A070518 Value of n-th cyclotomic polynomial at n.

0, 3, 13, 17, 781, 31, 137257, 4097, 532171, 9091, 28531167061, 20593, 25239592216021, 7027567, 2392743361, 4294967297, 51702516367896047761, 34006393, 109912203092239643840221, 25536159601, 7006306553612521, 25405143539623, 949112181811268728834319677753

Offset: 1

Labos Elemer, May 02 2002

nonn

a(28341) is divisible by 283411^2. What is the next n such that a(n) is not squarefree? - Jianing Song, Nov 01 2024

			n=10: 10th cyclotomic polynomial is 1-x+x^2-x^3+x^4; at x=10 it gives a(10)=9091.

G. C. Greubel, Table of n, a(n) for n = 1..250
Mathematics Stack Exchange, Is the "cyclotomic diagonalization" always squarefree?
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A070519 (indices of prime terms), A088790 (prime indices of prime terms).

a:= n-> numtheory[cyclotomic](n$2):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 05 2024

Mathematica
```
Table[Cyclotomic[w, w], {w, 1, 35}]
```

a(n) = polcyclo(n, n) \\ Michel Marcus, Apr 02 2016

A088790 Numbers k such that (k^k-1)/(k-1) is prime.

Original entry on oeis.org

2, 3, 19, 31, 7547

Offset: 1

T. D. Noe, Oct 16 2003

Note that (k^k-1)/(k-1) is prime only if k is prime, in which case it equals cyclotomic(k,k), the k-th cyclotomic polynomial evaluated at x=k. This sequence is a subsequence of A070519. The number cyclotomic(7547,7547) is a probable prime found by H. Lifchitz. Are there only a finite number of these primes?
From T. D. Noe, Dec 16 2008: (Start)
The standard heuristic implies that there are an infinite number of these primes and that the next k should be between 10^10 and 10^11.
Let N = (7547^7547-1)/(7547-1) = A023037(7547). If N is prime, then the period of the Bell numbers modulo 7547 is N. See A054767. (End)

R. K. Guy, Unsolved Problems in Theory of Numbers, 1994, A3.

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A070519 (cyclotomic(n, n) is prime).

Cf. A056826 ((n^n+1)/(n+1) is prime).

Do[p=Prime[n]; If[PrimeQ[(p^p-1)/(p-1)], Print[p]], {n, 100}]

is(n)=ispseudoprime((n^n-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

A070520 Prime numbers of form p=Cyclotomic(n,n).

Original entry on oeis.org

3, 13, 17, 31, 9091, 20593, 7027567, 109912203092239643840221, 568972471024107865287021434301977158534824481, 3726767253346131780312487317315864271, 19342489361037647362917398912701853907582504971824424988419390687587

Offset: 1

Labos Elemer, May 02 2002

nonn

Cf. A070518, A070519.

Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[s]], {n, 2, 256}]; output = the prime value
Select[Table[Cyclotomic[n,n],{n,300}],PrimeQ] (* Harvey P. Dale, Dec 06 2012 *)

A070523 Numbers k such that cyclotomic(k, prime(k)) is a prime number.

Original entry on oeis.org

3, 6, 7, 14, 19, 31, 34, 66, 93, 307, 402, 421, 600, 848, 1022, 1057, 1906, 3772, 4184, 4364

Offset: 1

Labos Elemer, May 02 2002

Values corresponding to a(3)=7 through a(13)=600 have been certified prime with Primo. Their sizes in decimal digits are 8, 10, 33, 64, 35, 51, 162, 1012, 455, 1455 and 584, respectively. Values corresponding to a(14) through a(17) are probable primes with lengths 1588, 1689, 3535 and 4014 decimal digits. a(18)>2000. - Rick L. Shepherd, Jul 10 2002

All terms <= 1906 have been proven with PARI's ECPP. No other terms <= 20000. - Lucas A. Brown, Jan 03 2021

			For n=7: 1+x+x^2+x^3+x^4+x^5+x^6 at x=prime(7)=17 gives a prime 25646167.

Lucas A. Brown, A070523.py.

Cf. A070518, A070519, A070520, A070521, A070522.

for(n=1,2000, if(isprime(eval(polcyclo(n, prime(n)))), print1(n, ", ")))

More terms from Rick L. Shepherd, Jul 10 2002

a(18)-a(20) by Lucas A. Brown, Jan 02 2021

A070525 Numbers n such that n-th cyclotomic polynomial evaluated at phi(n) is a prime number.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 12, 18, 21, 30, 45, 48, 70, 120, 127, 153, 182, 204, 212, 282, 318, 322, 910, 1167, 1177, 1342, 1680, 1963, 2670, 4398, 4655, 8088, 8599, 8808, 19680

Offset: 1

Labos Elemer, May 02 2002

These are probable primes for n > 910. No others for n <= 10000. The prime values of n are 2, 3, 7, 127 and 8599 (A088856). - T. D. Noe, Nov 23 2003

All terms <= 2670, except 1963, have been certified prime with PARI's ECPP. There are no other terms <= 25000. - Lucas A. Brown, Jan 08 2021

			n=7: Phi(7)=6, Cyclotomic(7,6)=1+6+36+216+1296+7776+46656=55987 is prime.

Lucas A. Brown, A070525.py.

Cf. A070518, A070519, A070520, A070521, A070522, A070523, A070524.

Cf. A088856, A090159.

Do[s=Cyclotomic[n, EulerPhi[n]]; If[PrimeQ[s], Print[n]], {n, 1, 400}]

isok(n) = isprime(polcyclo(n, eulerphi(n))); \\ Michel Marcus, Sep 01 2019

More terms from T. D. Noe, Nov 23 2003

a(35) by Lucas A. Brown, Jan 08 2021

A088817 Numbers k such that Cyclotomic(2k,k) is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 17, 36, 157, 245, 352, 3977

Offset: 1

T. D. Noe, Oct 20 2003

This is a generalization of A056826. Note that (n^n+1)/(n+1) = cyclotomic(2n,n) when n is prime. These are probable primes for n > 352. No others < 4700.

All terms of this sequence that are greater than 3 are congruent to 0 or 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 2, 3 (mod 4), then Cyclotomic(2k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Aurifeuillean factorization

Cf. A056826 ((k^k+1)/(k+1) is prime), A070519 (cyclotomic(k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Do[p=Prime[n]; If[PrimeQ[Cyclotomic[2n, n]], Print[p]], {n, 100}]

is(n)=ispseudoprime(polcyclo(2*n,n)) \\ Charles R Greathouse IV, May 22 2017

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A088856 Primes p such that cyclotomic(p,p-1) is prime.

Original entry on oeis.org

2, 3, 7, 127, 8599

Offset: 1

T. D. Noe, Nov 23 2003

Some of the larger entries may only correspond to probable primes.

For p > 2, these are numbers p such that ((p-1)^p - 1)/(p-2) is prime. - Thomas Ordowski, Sep 02 2021

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A070519, A070525, A101753.

lst={}; Do[p=Prime[n]; If[PrimeQ[Cyclotomic[p, p-1]], AppendTo[lst, p]], {n, 400}]; lst

isok(p) = isprime(p) && isprime(polcyclo(p, p-1)); \\ Michel Marcus, Sep 02 2021

a(n) = A101753(n) + 1. - Thomas Ordowski, Sep 02 2021

A088875 Cyclotomic(n,-n) is prime.

Original entry on oeis.org

1, 3, 4, 5, 6, 9, 12, 14, 17, 82, 86, 157, 158, 180, 210, 236, 245, 368, 462, 842, 1034, 3512, 3977, 8636

Offset: 1

T. D. Noe, Oct 20 2003

This is a generalization of A056826. See A088817 for another generalization. Note that (n^n+1)/(n+1) = cyclotomic(n,-n) when n is prime. Also note that, for odd n>1, cyclotomic(n,-n) = cyclotomic(2n,n) and for n a multiple of 4, cyclotomic(n,-n) = cyclotomic(n,n).

Some of the larger entries may only correspond to probable primes.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A056826 ((n^n+1)/(n+1) is prime), A070519 (cyclotomic(n, n) is prime), A088817 (cyclotomic(2n, n) is prime).

Do[p=Prime[n]; If[PrimeQ[Cyclotomic[n, -n]], Print[p]], {n, 100}]

cp's OEIS Frontend

A070518 Value of n-th cyclotomic polynomial at n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A088790 Numbers k such that (k^k-1)/(k-1) is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

A070520 Prime numbers of form p=Cyclotomic(n,n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A070523 Numbers k such that cyclotomic(k, prime(k)) is a prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A070525 Numbers n such that n-th cyclotomic polynomial evaluated at phi(n) is a prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A088817 Numbers k such that Cyclotomic(2k,k) is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A088856 Primes p such that cyclotomic(p,p-1) is prime.

Views

Author

Keywords