A045546 -id:A045546 - cp's OEIS frontend

A125881 Numbers k for which k^3+k^2-1 is prime.

2, 4, 5, 6, 9, 11, 12, 14, 19, 22, 25, 26, 27, 29, 32, 34, 36, 37, 40, 44, 47, 49, 55, 60, 62, 64, 65, 69, 70, 71, 81, 82, 84, 89, 95, 97, 106, 107, 114, 119, 121, 125, 127, 132, 139, 140, 141, 144, 147, 155, 159, 161, 165, 172, 174, 179, 184, 190, 201, 204

Offset: 1

Artur Jasinski, Dec 14 2006

nonn

Generating polynomial belongs to the family of irreducible polynomials (or trinomials) of the form x^n+x^(n-1)-1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A045546, A125882, A125883, A125884, A125885.

Do[If[PrimeQ[x^3 + x^2 - 1], Print[x]], {x, 1, 100}]
Select[Range[250], PrimeQ[#^3 + #^2 - 1] &] (* Harvey P. Dale, Jan 20 2015 *)

isok(n) = isprime(n^3+n^2-1); \\ Michel Marcus, Nov 08 2013

More terms from Michel Marcus, Nov 08 2013

A125885 Numbers k for which k^7+k^6-1 is prime.

Original entry on oeis.org

2, 4, 12, 17, 27, 39, 45, 46, 72, 94, 105, 106, 122, 126, 147, 149, 151, 156, 160, 161, 166, 169, 171, 172, 192, 194, 204, 205, 209, 230, 235, 242, 252, 260, 264, 266, 276, 280, 285, 306, 309, 319, 330, 335, 357, 371, 387, 400, 402, 411, 422, 439, 442, 451, 459

Offset: 1

Artur Jasinski, Dec 14 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A045546, A125881, A125882, A125883, A125884.

Do[If[PrimeQ[x^7 + x^6 - 1], Print[x]], {x, 1, 200}]

is(n)=isprime(n^7+n^6-1) \\ Charles R Greathouse IV, May 15 2013

More terms from Amiram Eldar, Mar 18 2020

A125965 Numbers k for which k^8+k^7-1 is prime.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 18, 23, 32, 36, 41, 52, 54, 55, 56, 60, 68, 74, 80, 82, 87, 93, 107, 115, 140, 142, 146, 154, 162, 165, 170, 189, 227, 238, 253, 262, 263, 269, 276, 285, 300, 304, 305, 306, 308, 310, 315, 317, 332, 339, 350, 361, 363, 367, 371, 384, 386, 390

Offset: 1

Artur Jasinski, Dec 14 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A045546, A125881-A125885, A125966-A125973.

Select[Range[500],PrimeQ[#^8+#^7-1]&]  (* Harvey P. Dale, Feb 23 2011 *)

select(ispseudoprime, vector(500,n,n^8+n^7-1)) \\ Charles R Greathouse IV, Feb 23 2011

A125973 Smallest k such that k^n + k^(n-1) - 1 is prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 2, 2, 14, 4, 7, 2, 38, 6, 7, 3, 4, 10, 2, 9, 74, 6, 10, 7, 4, 61, 20, 4, 5, 9, 6, 16, 6, 8, 2, 9, 4, 10, 2, 48, 44, 163, 9, 2, 95, 3, 27, 70, 6, 26, 57, 9, 6, 8, 207, 2, 27, 15, 45, 7, 69, 199, 55, 16, 2, 5, 12, 43, 137, 39, 9, 57, 5, 20, 4, 115, 2, 103, 45, 15, 20, 109

Offset: 1

Artur Jasinski, Dec 14 2006

nonn

The polynomial x^n + x^(n-1) - 1 is irreducible over the rationals (see Ljunggren link), so the Bunyakovsky conjecture implies that a(n) always exists. - Robert Israel, Nov 16 2016

			Consider n = 6. k^6 + k^5 - 1 evaluates to 1, 95, 971 for k = 1, 2, 3. Only the last of these numbers is prime, hence a(6) = 3.

Robert Israel, Table of n, a(n) for n = 1..800
W. Ljunggren, On the irreducibility of certain trinomials and quadrinomials, Math. Scand. 8 (1960) 65-70.
Wikipedia, Bunyakovsky conjecture.

Cf. A000040, A045546, A125881-A125885, A125965-A125972, A126017.

Cf. A091997 (n such that a(n)=2).

f:= proc(n) local k;
for k from 2 do if isprime(k^n+k^(n-1)-1) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 16 2016

a[n_] := For[k = 2, True, k++, If[PrimeQ[k^n + k^(n-1) - 1], Return[k]]];
Array[a, 100] (* Jean-François Alcover, Feb 26 2019 *)

{m=82;for(n=1,m,k=1;while(!isprime(k^n+k^(n-1)-1),k++);print1(k,","))} \\ Klaus Brockhaus, Dec 17 2006

Edited and extended by Klaus Brockhaus, Dec 17 2006

A125964 Numbers k such that k^3 + k^2 + k - 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 18, 20, 24, 34, 46, 50, 56, 58, 70, 72, 78, 84, 92, 108, 112, 116, 142, 146, 150, 168, 172, 176, 178, 186, 188, 190, 204, 230, 238, 240, 242, 244, 252, 270, 276, 284, 286, 296, 304, 310, 328, 342, 350, 352, 354, 370, 378, 384, 388, 400, 406

Offset: 1

Artur Jasinski, Dec 13 2006

nonn

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

Cf. A045546, A125965-A125973.

[n: n in [0..1000] | IsPrime(n^3+n^2+n-1)] // Vincenzo Librandi, Nov 23 2010

filter:= k -> isprime(k^3+k^2+k-1):
select(filter, [1,seq(i,i=2..10000,2)]); # Robert Israel, Oct 07 2019

Do[If[PrimeQ[ -1+n+n^2+n^3],Print[n]],{n,1,300}]

is(n)=isprime(n^3+n^2+n-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Vincenzo Librandi, Mar 26 2010

A125882 Numbers k for which k^4+k^3-1 is prime.

Original entry on oeis.org

2, 3, 6, 11, 13, 18, 24, 34, 38, 39, 43, 49, 52, 57, 58, 73, 74, 79, 90, 102, 104, 107, 113, 116, 122, 123, 126, 135, 139, 148, 155, 181, 183, 188, 193, 199, 203, 223, 234, 240, 247, 256, 261, 272, 273, 277, 286, 288, 298, 303, 329, 338, 344, 346, 357, 364, 366

Offset: 1

Artur Jasinski, Dec 14 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A045546, A125881, A125883, A125884, A125885.

Do[If[PrimeQ[x^4 + x^3 - 1], Print[x]], {x, 1, 100}]

is(n)=isprime(n^4+n^3-1) \\ Charles R Greathouse IV, May 15 2013

More terms from Amiram Eldar, Mar 18 2020

A125883 Numbers k for which k^5+k^4-1 is prime.

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 16, 17, 18, 26, 27, 29, 31, 33, 34, 38, 39, 40, 43, 47, 50, 57, 59, 65, 69, 70, 76, 81, 89, 90, 92, 93, 95, 103, 107, 109, 126, 128, 129, 138, 140, 148, 162, 167, 179, 182, 183, 192, 197, 206, 209, 211, 221, 223, 226, 228, 229, 230, 234, 240

Offset: 1

Artur Jasinski, Dec 14 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A045546, A125881, A125882, A125884, A125885.

Do[If[PrimeQ[x^5 + x^4 - 1], Print[x]], {x, 1, 200}]
Select[Range[200],PrimeQ[#^5+#^4-1]&] (* Harvey P. Dale, Apr 10 2018 *)

is(n)=isprime(n^5+n^4-1) \\ Charles R Greathouse IV, May 15 2013

More terms from Amiram Eldar, Mar 18 2020

A125884 Numbers k for which k^6+k^5-1 is prime.

Original entry on oeis.org

3, 4, 5, 8, 9, 16, 18, 30, 34, 36, 41, 48, 49, 51, 54, 61, 65, 69, 76, 81, 86, 89, 90, 95, 101, 109, 120, 141, 170, 171, 178, 196, 221, 235, 238, 244, 260, 263, 273, 280, 291, 301, 303, 310, 311, 326, 350, 361, 375, 391, 398, 404, 405, 406, 423, 429, 431, 454, 456, 464, 479, 484, 485, 486, 489, 499

Offset: 1

Artur Jasinski, Dec 14 2006

nonn

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

Cf. A000040, A045546, A125881, A125882, A125883, A125885.

select(t -> isprime(t^6+t^5-1), [$1..1000]); # Robert Israel, Jan 08 2017

Do[If[PrimeQ[x^6 + x^5 - 1], Print[x]], {x, 1, 200}]

is(n)=isprime(n^6+n^5-1) \\ Charles R Greathouse IV, May 15 2013

More terms from Robert Israel, Jan 08 2017

A237615 a(n) = |{0 < k < n: k^2 + k - 1 and pi(k*n) are both prime}|, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 4, 1, 3, 4, 4, 2, 4, 3, 6, 2, 2, 2, 3, 7, 4, 3, 4, 5, 6, 1, 3, 2, 3, 9, 3, 3, 4, 7, 5, 8, 5, 2, 2, 5, 5, 4, 5, 6, 4, 5, 6, 10, 6, 6, 10, 9, 9, 10, 12, 2, 8, 7, 3, 6, 6, 4, 6

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) For each n = 4, 5, ..., there is a positive integer k < n with k^2 + k - 1 and pi(k*n) + 1 both prime. Also, for any integer n > 6, there is a positive integer k < n with k^2 + k - 1 and pi(k*n) - 1 both prime.
(iii) For every integer n > 15, there is a positive integer k < n such that pi(k) - 1 and pi(k*n) are both prime.
Note that part (i) is a refinement of the first assertion in the comments in A237578.

			a(8) = 1 since 4^2 + 4 - 1 = 19 and pi(4*8) = 11 are both prime.
a(33) = 1 since 28^2 + 28 - 1 = 811 and pi(28*33) = 157 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, A combinatorial conjecture on primes, a message to Number Theory List, Feb. 9, 2014.

Cf. A000040, A000720, A002327, A045546, A237578, A237597, A237598.

p[k_,n_]:=PrimeQ[k^2+k-1]&&PrimeQ[PrimePi[k*n]]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n-1}]
Table[a[n],{n,1,70}]

A239135 Numbers k such that (k-1)*k^2 + 1 and k^2 + (k-1) are both prime.

Original entry on oeis.org

2, 3, 5, 6, 8, 13, 21, 24, 26, 28, 35, 45, 48, 50, 55, 76, 83, 89, 93, 96, 100, 101, 115, 120, 138, 140, 148, 149, 181, 191, 195, 203, 206, 209, 215, 230, 258, 259, 281, 285, 294, 301, 309, 323, 330, 349, 358, 373, 380, 386, 393, 395, 423, 428, 433, 474, 495

Offset: 1

Ilya Lopatin following a suggestion from Juri-Stepan Gerasimov, Mar 15 2014

nonn

Numbers k such that (k^3 - k^2 + 1)*(k^2 + k - 1) is semiprime.
Intersection of A045546 and A111501.
Primes in this sequence: 2, 3, 5, 13, 83, 89, 101, 149, 181, 191, ...

			2 is in this sequence because (2-1)*2^2+1=5 and 2^2+(2-1)=5 are both prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A239115.

k := 1;
     for n in [1..10000] do
        if IsPrime(k*(n - 1)*n^2 + 1) and IsPrime(k*n^2 + n - 1) then
           n;
        end if;
     end for;

Select[Range[600],PrimeQ[#^2+#-1]&&PrimeQ[#^2(#-1)+1]&] (* Farideh Firoozbakht, Mar 17 2014 *)

cp's OEIS Frontend

A125881 Numbers k for which k^3+k^2-1 is prime.

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A125885 Numbers k for which k^7+k^6-1 is prime.

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A125965 Numbers k for which k^8+k^7-1 is prime.

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A125973 Smallest k such that k^n + k^(n-1) - 1 is prime.

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A125964 Numbers k such that k^3 + k^2 + k - 1 is prime.

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A125882 Numbers k for which k^4+k^3-1 is prime.

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A125883 Numbers k for which k^5+k^4-1 is prime.

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A125884 Numbers k for which k^6+k^5-1 is prime.

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