A060891 -id:A060891 - cp's OEIS frontend

A269442 a(n) = n(n^8 + 1)(n^4 + 1)(n^2 + 1)(n + 1) + 1.

1, 17, 131071, 64570081, 5726623061, 190734863281, 3385331888947, 38771752331201, 321685687669321, 2084647712458321, 11111111111111111, 50544702849929377, 201691918794585181, 720867993281778161, 2345488209948553531, 7037580381120954241

Offset: 0

Ilya Gutkovskiy, Feb 26 2016

a(n) = Phi_17(n) where Phi_k(x) is the k-th cyclotomic polynomial.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. similar sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), this sequence (k=17), A060891 (k=18), A269446 (k=19).

List([0..20], n-> n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1); # G. C. Greubel, Apr 24 2019

[n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1: n in [0..20]]; // Vincenzo Librandi, Feb 27 2016

Mathematica
```
Table[Cyclotomic[17, n], {n, 0, 15}]
```

a(n)=n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1 \\ Charles R Greathouse IV, Jul 26 2016

[n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1 for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 +130918*x^2 +62343506*x^3 +4646748160*x^4 +102074708252*x^5 +878064150546*x^6 +3419813860214*x^7 +6502752956958*x^8 +6232856389160*x^9 +3004612851498*x^10 +701875014878*x^11 +73106078368*x^12 +2893069436*x^13 +31542430*x^14 +43674*x^15 +x^16)/(1 - x)^17.

Sum_{n>=0} 1/a(n) = 1.05883117453...

A175170 Primes of the form k^6 - k^3 + 1.

Original entry on oeis.org

46441, 117307, 530713, 5488957657, 13841169553, 24794753833, 82653662521, 151333837273, 282429005041, 464403405313, 782756904961, 999999000001, 1418517921241, 2699552509993, 2985982272001, 3138426605161, 4001502141001, 28722895031251, 47045874141001, 56693904845761, 78672332016307

Offset: 1

Vincenzo Librandi, Dec 03 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A060891, A164989.

[ a: n in [0..250] | IsPrime(a) where a is n^3*(n^3-1)+1 ];

Select[Table[n^6-n^3+1,{n,0,2000}],PrimeQ] (* Vincenzo Librandi, Jul 26 2012 *)

A000040 INTERSECT A060891.

A175205 Primes p such that p^6-p^3+1 is also prime.

Original entry on oeis.org

7, 73, 271, 367, 379, 397, 739, 823, 859, 1291, 1297, 1627, 1747, 2083, 2203, 2221, 2269, 2311, 3067, 3079, 3109, 3163, 3463, 3607, 3733, 3907, 4093, 4339, 4603, 4933, 5077, 5431, 5527, 5821, 5869, 6091, 6301, 6373, 6421, 6529, 6883, 7927, 7951, 8011, 8467, 8689, 8737

Offset: 1

Vincenzo Librandi, Dec 03 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A060891, A175170, A000040, A164989.

Subsequence of A002476.

[p: p in PrimesUpTo(9000) | IsPrime(p^6-p^3+1)]; // Vincenzo Librandi, Apr 16 2013

Select[Prime[Range[2000]], PrimeQ[#^6 -  #^3 + 1]&] (* Vincenzo Librandi, Apr 16 2013 *)

forprime(p=2,10000,if(isprime(p^6-p^3+1),print1(p,", ")))

A000040 INTERSECT A164989.

A326618 a(n) = n^18 + n^9 + 1.

Original entry on oeis.org

1, 3, 262657, 387440173, 68719738881, 3814699218751, 101559966746113, 1628413638264057, 18014398643699713, 150094635684419611, 1000000001000000001, 5559917315850179173, 26623333286045024257, 112455406962561892503, 426878854231297789441, 1477891880073843750001

Offset: 0

Richard N. Smith, Jul 15 2019

nonn

a(n) = Phi_27(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Richard N. Smith, Table of n, a(n) for n = 0..2000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument

Sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A269442 (k=17), A060891 (k=18), A269446 (k=19), A060892 (k=20), A269483 (k=21), A269486 (k=22), A060893 (k=24), A269527 (k=25), A266229 (k=26), this sequence (k=27), A270204 (k=28), A060894 (k=30), A060895 (k=32), A060896 (k=36).

Cf. A153440 (indices of prime terms).

[n^18+n^9+1: n in [0..17]]; // Vincenzo Librandi, Jul 15 2019

Table[n^18 + n^9 + 1, {n, 0, 17}] (* Vincenzo Librandi, Jul 15 2019 *)
Table[Cyclotomic[27, n], {n, 0, 17}]

a(n) = polcyclo(27, n); \\ Michel Marcus, Jul 20 2019

cp's OEIS Frontend

A269442 a(n) = n*(n^8 + 1)*(n^4 + 1)*(n^2 + 1)*(n + 1) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A175170 Primes of the form k^6 - k^3 + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A175205 Primes p such that p^6-p^3+1 is also prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A326618 a(n) = n^18 + n^9 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A269442 a(n) = n(n^8 + 1)(n^4 + 1)(n^2 + 1)(n + 1) + 1.