A070518 -id:A070518 - cp's OEIS frontend

A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

2, 3, 4, 6, 10, 12, 14, 19, 31, 46, 74, 75, 98, 102, 126, 180, 236, 310, 368, 1770, 1858, 3512, 4878, 5730, 7547, 7990, 8636, 9378, 11262

Offset: 1

Labos Elemer, May 02 2002

When n is prime, then the solutions are given in A088790.
No term of this sequence is congruent to 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 1 (mod 4), then Cyclotomic(k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019
All terms <= 1858 have been proven with PARI's implementation of ECPP.  All larger terms are BPSW PRPs.  There are no further terms <= 30000. - Lucas A. Brown, Dec 28 2020

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

Cf. A070518, A070520, A088790 ((k^k-1)/(k-1) is prime), A088817 (cyclotomic(2k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Cf. A117544, A085398.

Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[n]], {n, 2, 256}]

for(n=2,10^9,if(ispseudoprime(polcyclo(n,n)),print1(n,", "))); \\ Joerg Arndt, Jan 22 2015

More terms from T. D. Noe, Oct 17 2003

a(29) from Charles R Greathouse IV, May 05 2011

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 *)

A070521 Value of prime(n)-th cyclotomic polynomial at n.

Original entry on oeis.org

2, 7, 121, 5461, 12207031, 2612138803, 38771752331201, 20587884010836553, 1107867264956562636991, 11111111111111111111111111111, 19194342495775048050414684129181, 773238409074645649506921062192257266781, 391287702091575733090747617444785169589677401

Offset: 1

Labos Elemer, May 02 2002

nonn

			n=3: Cyclotomic[prime(3),x]=C[5,3]=[1+x+xx+xxx+xxxx,at x=3]; it is 1+3+9+27+81=121=a(3).

Cf. A070518-A070520.

Table[Cyclotomic[Prime[w], w], {w, 1, 25}]

a(n) = polcyclo(prime(n), n); \\ Michel Marcus, Aug 29 2019

a(n) = Cyclotomic[Prime[n], n]

More terms from Michel Marcus, Aug 29 2019

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

A070526 Value of n-th cyclotomic polynomial at 2^n.

Original entry on oeis.org

1, 5, 73, 257, 1082401, 4033, 4432676798593, 4294967297, 18014398643699713, 1098438933505, 1298708349570020393652962442872833, 281474959933441, 91355004067076339167413824240109498970069278721

Offset: 1

Labos Elemer, May 02 2002

			n=5: a(5)=C[5,32]=1+32+1024+32768+1048576=1082401

Cf. A070518-A070525.

Mathematica
```
Table[Cyclotomic[w, 2^w], {w, 1, 15}]
```

A138919 Indices k such that A020510(k)=Phi[k](-11) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

5, 7, 18, 20, 21, 30, 34, 36, 38, 52, 54, 60, 90, 104, 117, 123, 146, 159, 179, 182, 229, 278, 388, 405, 410, 439, 552, 557, 735, 806, 807, 1220, 1272, 1568, 1688, 1696, 1710, 1814, 2136, 2262, 2288, 2862, 3679, 3814, 4058, 4304, 4480, 5070, 5136, 5154

Offset: 1

M. F. Hasler, Apr 04 2008

nonn

Larger values are probable primes.

Robert Price, Table of n, a(n) for n = 1..91
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A020510, A138920-A138940, A070518-A070527.

Select[Range[1000], PrimeQ[Cyclotomic[#, -11]] &]

for( i=1,999, ispseudoprime( polcyclo(i,-11)) & print1(",",i)) /* use ...subst(polcyclo(i),x,-11)... in PARI < 2.4.2 */

Edited by T. D. Noe, Oct 30 2008
a(32)-a(44) from Robert Price, Mar 16 2012
a(45)-a(50) from Robert Price, Apr 14 2012

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)

A070522 Value of n-th cyclotomic polynomial at the n-th prime.

Original entry on oeis.org

1, 4, 31, 50, 16105, 157, 25646167, 130322, 148048057, 683705, 846949229880161, 1872793, 23127577557875340733, 6177695707, 23304888110401, 62259690411362, 21930887362370823132822661921, 51520147381

Offset: 1

Labos Elemer, May 02 2002

nonn

			n=5: Cyclotomic[5,x]=1+x+xx+xxx+xxxx; at x=p(5)=11 it gives a(5)=16105.

Cf. A070518-A070521.

Table[Cyclotomic[w, Prime[w]], {w, 1, 20}]

a(n)=Cyclotomic[n, Prime[n]]

A070527 Numbers k such that cyclotomic(k, 2^k) is a prime number.

Original entry on oeis.org

2, 3, 4, 7, 40, 59, 140

Offset: 1

Labos Elemer, May 02 2002

No other terms below 300. - Max Alekseyev, May 29 2009

No other terms below 760. - Lucas A. Brown, Dec 16 2020

			4 is a term because cyclotomic(4,16) = 1 + 16^2 = 257 is prime.

Cf. A070518-A070526.

Do[s=Cyclotomic[n, 2^n]; If[PrimeQ[s], Print[n]], {n, 1, 1000}]
Select[Range[150],PrimeQ[Cyclotomic[#,2^#]]&] (* Harvey P. Dale, Aug 12 2023 *)

isok(k) = isprime(polcyclo(k, 2^k)); \\ Michel Marcus, Dec 17 2020

One more term from Max Alekseyev, May 29 2009

cp's OEIS Frontend

A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

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

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A070521 Value of prime(n)-th cyclotomic polynomial at n.

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

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

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A070526 Value of n-th cyclotomic polynomial at 2^n.

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A138919 Indices k such that A020510(k)=Phi[k](-11) is prime, where Phi is a cyclotomic polynomial.

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Mathematica

PARI

Extensions

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

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