A125082 -id:A125082 - cp's OEIS frontend

A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 14, 1, 1, 5, 19, 43, 41, 1, 1, 6, 29, 94, 171, 122, 1, 1, 7, 41, 173, 469, 683, 365, 1, 1, 8, 55, 286, 1037, 2344, 2731, 1094, 1, 1, 9, 71, 439, 2001, 6221, 11719, 10923, 3281, 1, 1, 10, 89, 638, 3511, 14006, 37325, 58594, 43691, 9842, 1

Offset: 0

Paul Barry, Apr 21 2003

			Rows begin:
1  1   1    1     1      1       1        1         1 ...
1  2   5   14    41    122     365     1094      3281 ...  A007051
1  3  11   43   171    683    2731    10923     43691 ...  A007583
1  4  19   94   469   2344   11719    58594    292969 ...  A083065
1  5  29  173  1037   6221   37325   223949   1343693 ...  A083066
1  6  41  286  2001  14006   98041   686286   4804001 ...  A083067
1  7  55  439  3511  28087  224695  1797559  14380471 ...  A083068
1  8  71  638  5741  51668  465011  4185098  37665881 ...  A187709
1  9  89  889  8889  88889  888889  8888889  88888889 ...  A059482
1 10 109 1198 13177 144946 1594405 17538454 192922993 ...  A199760, etc.
Column 2: A000027;
column 3: A028387;
column 4: A083074;
column 5: A125082;
column 6: A125083.
Diagonals:
1,  2,  11,   94,  1037,  14006, ... A083069;
1,  3,  19,  173,  2001,  28087, ... A083071;
1,  4,  29,  286,  3511,  51668, ... A083072;
1,  5,  41,  439,  5741,  88889, ... A083073;
1,  5,  43,  469,  6221,  98041, ... A083070;
1, 14, 171, 2344, 37325, 686286, ... A191690.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 5, 1;
1, 4, 11, 14, 1;
1, 5, 19, 43, 41, 1;
1, 6, 29, 94, 171, 122, 1; etc.

Cf. rows: A007051, A007583, A059482, A083065 - A083068, A187709, A199760; columns: A000027, A028387, A083074, A125082, A125083; diagonals: A083069 - A083073, A191690.

Edited by Bruno Berselli, Jun 21 2013

A125083 a(n) = n^5-n^4-n^3-n^2-n-1.

Original entry on oeis.org

-1, -4, 1, 122, 683, 2344, 6221, 14006, 28087, 51668, 88889, 144946, 226211, 340352, 496453, 705134, 978671, 1331116, 1778417, 2338538, 3031579, 3879896, 4908221, 6143782, 7616423, 9358724, 11406121, 13797026, 16572947, 19778608, 23462069, 27674846, 32472031, 37912412, 44058593

Offset: 0

Artur Jasinski, Nov 19 2006

More generally, the ordinary generating function for the values of quintic polynomial b*n^5 + p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (b + p + q + k + m - 5*r)*x + (13*b + 5*p + q - k - 2*m + 5*r)*2*x^2 + (33*b - 3*q + 3*m - 5*r)*2*x^3 + (26*b - 10*p + 2*q + 2*k - 4*m + 5*r)*x^4 + (b - p + q - k + m - r)*x^5)/(1 - x)^6. - Ilya Gutkovskiy, Mar 31 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..580
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A125082, A083074.

[n^5-n^4-n^3-n^2-n-1: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

Table[n^5 - n^4 - n^3 - n^2 - n - 1, {n, 0, 41}]

a(n) = n^5-n^4-n^3-n^2-n-1; \\ Michel Marcus, Mar 31 2016

G.f.: (-1 + 2*x + 10*x^2 + 76*x^3 + 31*x^4 + 2*x^5)/(1 - x)^6. - Ilya Gutkovskiy, Mar 31 2016

A237639 Numbers n = p^4-p^3-p^2-p-1 (for prime p) such that n^4-n^3-n^2-n-1 is prime.

Original entry on oeis.org

41, 56133395601, 89362058601, 590884122501, 1275627652881, 2775672202617, 6212311361721, 7534036143501, 27344792789601, 61180709716101, 124857759197601, 206926840439901, 580608824590341, 603653936046501, 1442441423278281, 1864059458505657

Offset: 1

Derek Orr, Feb 10 2014

nonn

All numbers are congruent to 1 mod 10 or 7 mod 10.

41 is the only prime in the sequence, since one of p, n, and n^4-n^3-n^2-n-1 must be divisible by 3. - Charles R Greathouse IV, Feb 11 2014

			41 = 3^4-3^3-3^2-3^1-1 (3 is prime) and 41^4-41^3-41^2-41^1-1 = 2755117 is prime. So, 41 is a member of this sequence.

Cf. A125082.

s=[]; forprime(p=2, 7000, n=p^4-p^3-p^2-p-1; if(isprime(n^4-n^3-n^2-n-1), s=concat(s, n))); s \\ Colin Barker, Feb 11 2014

import sympy
from sympy import isprime
def poly4(x):
  if isprime(x):
    f = x**4-x**3-x**2-x-1
    if isprime(f**4-f**3-f**2-f-1):
      return True
  return False
x = 1
while x < 10**5:
  if poly4(x):
    print(x**4-x**3-x**2-x-1)
  x += 1

A173179 Numbers n such that n^4-n^3-n^2-n-1 is prime.

Original entry on oeis.org

3, 8, 9, 11, 14, 17, 18, 20, 24, 27, 38, 41, 45, 48, 50, 51, 56, 59, 60, 62, 63, 71, 77, 78, 81, 84, 86, 87, 90, 92, 93, 95, 101, 111, 113, 114, 119, 146, 147, 153, 155, 171, 179, 186, 204, 207, 219, 225, 230, 231, 233, 234, 240, 246, 254, 255, 267, 284, 287, 291

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

All terms == 0 or 2 (mod 3). - Robert Israel, Mar 09 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A125082, A002328, A162294.

[n: n in [2..350] | IsPrime(n^4 - n^3 - n^2 - n - 1)]; // Vincenzo Librandi, Mar 16 2020

select(t -> isprime(t^4-t^3-t^2-t-1), [$2..1000]); # Robert Israel, Mar 09 2020

f[n_]:=n^4-n^3-n^2-n-1;Select[Range[6! ],PrimeQ[f[ #1]]&]

is(n)=isprime(n^4-n^3-n^2-n-1) \\ Charles R Greathouse IV, Jun 06 2017

Edited by N. J. A. Sloane, Apr 10 2010

A230029 Primes p such that f(f(p)) is prime, where f(x) = x^4-x^3-x^2-x-1.

Original entry on oeis.org

3, 487, 547, 877, 1063, 1291, 1579, 1657, 2287, 2797, 3343, 3793, 4909, 4957, 6163, 6571, 7393, 8461, 8521, 8563, 9631, 11257, 11863, 12211, 12757, 12907, 13063, 13567, 13999, 14983, 15427, 15739, 16087, 16651, 16699, 17419, 17713, 17977

Offset: 1

Derek Orr, Feb 23 2014

nonn

			3 is prime and (3^4-3^3-3^2-3-1)^4 - (3^4-3^3-3^2-3-1)^3 - (3^4-3^3-3^2-3-1)^2 - (3^4-3^3-3^2-3-1) - 1 = 2755117 is prime. Thus, 3 is a member of this sequence.

Cf. A000040, A237639, A125082.

from sympy import isprime
def f(x):
  return x**4-x**3-x**2-x-1
{print(p) for p in range(10**5) if isprime(p) and isprime(f(f(p)))}

cp's OEIS Frontend

A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

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A125083 a(n) = n^5-n^4-n^3-n^2-n-1.

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A237639 Numbers n = p^4-p^3-p^2-p-1 (for prime p) such that n^4-n^3-n^2-n-1 is prime.

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A173179 Numbers n such that n^4-n^3-n^2-n-1 is prime.

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Author

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Programs

Magma

Maple

Mathematica

PARI

Extensions

A230029 Primes p such that f(f(p)) is prime, where f(x) = x^4-x^3-x^2-x-1.

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Author

Keywords

Examples

Crossrefs

Programs

Python