cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, May 02 2002

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[n]], {n, 2, 256}]
  • PARI
    for(n=2,10^9,if(ispseudoprime(polcyclo(n,n)),print1(n,", "))); \\ Joerg Arndt, Jan 22 2015

Extensions

More terms from T. D. Noe, Oct 17 2003
a(29) from Charles R Greathouse IV, May 05 2011

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

Views

Author

Labos Elemer, May 02 2002

Keywords

Examples

			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).

Crossrefs

Cf. A070518-A070520.

Programs

  • Mathematica
    Table[Cyclotomic[Prime[w], w], {w, 1, 25}]
  • PARI
    a(n) = polcyclo(prime(n), n); \\ Michel Marcus, Aug 29 2019

Formula

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

Extensions

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

Views

Author

Labos Elemer, May 02 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Labos Elemer, May 02 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A070518, A070519, A070520, A070521, A070522, A070523, A070524.
Cf. A088856, A090159.

Programs

  • Mathematica
    Do[s=Cyclotomic[n, EulerPhi[n]]; If[PrimeQ[s], Print[n]], {n, 1, 400}]
  • PARI
    isok(n) = isprime(polcyclo(n, eulerphi(n))); \\ Michel Marcus, Sep 01 2019

Extensions

More terms from T. D. Noe, Nov 23 2003
a(35) by Lucas A. Brown, Jan 08 2021
Showing 1-4 of 4 results.

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