A002327 -id:A002327 - cp's OEIS frontend

A088502 Numbers n such that (n^2 - 5)/4 is prime.

5, 7, 9, 11, 13, 17, 19, 21, 23, 27, 31, 33, 39, 41, 43, 49, 53, 57, 61, 63, 71, 77, 79, 83, 89, 91, 93, 97, 101, 107, 109, 111, 113, 119, 121, 129, 131, 133, 137, 141, 153, 167, 171, 173, 179, 187, 189, 193, 201, 203, 207, 229, 231, 241, 251, 253, 261, 263, 269

Offset: 1

Pierre CAMI, Nov 13 2003

Under Bunyakovsky's conjecture this sequence is infinite. - Charles R Greathouse IV, Dec 28 2011

			(23*23 - 5)/4 = 131, 131 is prime, 23 is the 9th n of the sequence.

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

Cf. A002327, A002328, A110013.

[n: n in [1..300 by 2] | IsPrime((n^2-5) div 4)]; // Vincenzo Librandi, Oct 06 2012

Select[Range[500], PrimeQ[(#^2 - 5)/4] &] (* Vincenzo Librandi, Oct 06 2012 *)

for(k=2,1e3,if(isprime(k^2+k-1),print1(2*k+1", "))) \\ Charles R Greathouse IV, Dec 28 2011

a(n) = 2*A002328(n) - 1 = Sqrt(A110013(n)). - Ray Chandler, Sep 07 2005

A078179 a(n) is the smallest prime of the form n^k + n - 1 with k >= 2.

Original entry on oeis.org

5, 11, 19, 29, 41, 349, 71, 89, 109, 131, 20747, 181, 2177953337809371149, 239, 271, 83537, 5849, 379, 419, 461, 2494357909, 279863, 599, 15649, 701, 19709, 811, 420707233300229, 929, 991

Offset: 2

Reinhard Zumkeller, Nov 20 2002

nonn

			349 = A000040(70) = 7^3+7-1 and 7^2+7-1 = 5*11, therefore a(7) = 349.

Reinhard Zumkeller, Table of n, a(n) for n = 2..100

Cf. A078178, A076846.

Cf. A010051, A002327.

a078179 n = n ^ (a078178 n) + n - 1  -- Reinhard Zumkeller, Jul 16 2014

a(n) = n^A078178(n) + n - 1.

More terms from Benoit Cloitre, Nov 20 2002

Offset corrected by Reinhard Zumkeller, Jul 16 2014

A126435 Primes of the form n^7-n-1.

Original entry on oeis.org

2097143, 1801088519, 21869999969, 42618442943, 78364164059, 137231006639, 194754273839, 435817657169, 678223072799, 1174711139783, 1727094849479, 3938980639103, 4398046511039, 4902227890559, 6722988818363, 19203908986079

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

All terms end in 3 or 9. - Robert Israel, Jul 22 2019

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

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434.

map(t -> t^7-t-1, select(t -> isprime(t^7-t-1), [$1..10^4])); # Robert Israel, Jul 22 2019

k = 7; a = {}; Do[If[PrimeQ[x^k - x - 1], AppendTo[a, x^k - x - 1]], {x, 1, 100}]; a
Select[Table[n^7-n-1,{n,80}],PrimeQ] (* Harvey P. Dale, Jun 20 2020 *)

A126437 Primes of the form k^8-k-1.

Original entry on oeis.org

1679609, 5764793, 99999989, 4294967279, 282429536453, 377801998307, 5352009260441, 16815125390579, 39062499999949, 72301961339081, 83733937890569, 281474976710591, 513798374428571, 1113034787454899

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

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

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434, A126434.

k = 8; a = {}; Do[If[PrimeQ[x^k - x - 1], AppendTo[a, x^k - x - 1]], {x, 1, 100}]; a
Select[Table[k^8-k-1,{k,80}],PrimeQ] (* Harvey P. Dale, Nov 06 2021 *)

A140869 Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1.

Original entry on oeis.org

1, 2, 5, 4, 7, 11, 5, 10, 14, 19, 7, 12, 18, 23, 29, 8, 15, 21, 28, 34, 41, 10, 17, 25, 32, 40, 47, 55, 11, 20, 28, 37, 45, 54, 62, 71, 13, 22, 32, 41, 51, 60, 70, 79, 89, 14, 25, 35, 46, 56, 67, 77, 88, 98, 109, 16, 27, 39, 50, 62, 73, 85, 96, 108, 119, 131, 17, 30, 42, 55, 67, 80, 92, 105, 117, 130, 142, 155

Offset: 1

Vincenzo Librandi, Jan 16 2009

Conjecture: If h does not belong to the sequence, then 4*h+5 is prime. - Vincenzo Librandi, Nov 18 2012

First column: A001651; second column: A047215; third column: A047345. - Vincenzo Librandi, Nov 18 2012

			Triangle begins:
1;
2,  5;
4,  7,  11;
5,  10, 14, 19;
7,  12, 18, 23, 29;
8,  15, 21, 28, 34, 41;
10, 17, 25, 32, 40, 47, 55; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A002327, A153085, A001651, A047215, A047345.

[Floor(((2*n*k+n+k-2)/2)): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 18 2012

Flatten[Table[Floor[(2*n*m+m+n-2)/2],{n,1,10},{m,n}]] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

A237527 Numbers n of the form p^2-p-1 = A165900(p), for prime p, such that n^2-n-1 = A165900(n) is also prime.

Original entry on oeis.org

5, 155, 505, 2755, 3421, 6805, 11341, 27721, 29755, 31861, 44309, 49505, 52211, 65791, 100171, 121451, 134321, 185329, 195805, 236681, 252505, 258571, 292139, 325469, 375155, 380071, 452255, 457651, 465805, 563249, 676505, 1041419, 1061929

Offset: 1

Derek Orr, Feb 09 2014

nonn

All numbers are congruent to 1 mod 10, 5 mod 10, or 9 mod 10.

A subsequence of A165900 and A028387. - M. F. Hasler, Mar 01 2014

			5 = 3^2-3-1 (3 is prime) and 5^2-5-1 = 19 is also prime. So, 5 is a member of this sequence.

Cf. A237360, A039914, A002327.

s=[]; forprime(p=2, 40000, n=p^2-p-1; if(isprime(n^2-n-1), s=concat(s, n))); s \\ Colin Barker, Feb 10 2014

import sympy
from sympy import isprime
{print(n**2-n-1) for n in range(10**4) if isprime(n) and isprime((n**2-n-1)**2-(n**2-n-1)-1)}

a(n) = A165900(A230026(n)). - M. F. Hasler, Mar 01 2014

A126438 Primes of the form n^9-n-1.

Original entry on oeis.org

509, 262139, 10077689, 387420479, 68719476719, 118587876479, 1207269217769, 7625597484959, 10578455953379, 129961739795039, 327381934393919, 1628413597910399, 1953124999999949, 5416169448144839, 10077695999999939

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434, A126435, A126436, A126437.

k = 9; a = {}; Do[If[PrimeQ[x^k - x - 1], AppendTo[a, x^k - x - 1]], {x, 1, 100}]; a
Select[Table[n^9-n-1,{n,100}],PrimeQ] (* Harvey P. Dale, Mar 09 2016 *)

A171771 Primes of form n^6-(n+1)^5.

Original entry on oeis.org

971, 431441, 838949, 2614691, 6770161, 43845881, 570523321, 9244951889, 33640090481, 41402933641, 81303824909, 126165366289, 137240997911, 346860978491, 372445245449, 525200678549, 726938163649, 774170449439

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 18 2009

nonn

(1) It is conjectured that sequence is infinite.

(2) p=97=prime(25) is the smallest prime such that (p-1)^6-p^5 and p^6-(p+1)^5 are primes.

			4^6-5^5=971 and 9^6-10^5=431441 are prime.

Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Derrick H. Lehmer, Guide to Tables in the Theory of Numbers Washington, D.C. 1941

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002327, A140719, A087191

Select[Table[n^6-(n+1)^5,{n,3,100}],PrimeQ] (* Harvey P. Dale, Mar 06 2019 *)

Edited by D. S. McNeil, Nov 21 2010

A230026 Primes p such that f(f(p)) is prime, where f(n) = n^2-n-1 = A165900(n).

Original entry on oeis.org

3, 13, 23, 53, 59, 83, 107, 167, 173, 179, 211, 223, 229, 257, 317, 349, 367, 431, 443, 487, 503, 509, 541, 571, 613, 617, 673, 677, 683, 751, 823, 1021, 1031, 1093, 1103, 1109, 1123, 1201, 1231, 1289, 1301, 1319, 1361, 1373, 1427, 1451

Offset: 1

Derek Orr, Feb 23 2014

nonn

Note that f(f(f(n))) = (-1 + 4*n - 3*n^3 + n^4)*(1 + n - 3*n^2 - n^3 + n^4) is always composite. - Zak Seidov, Nov 10 2014

			3 is prime and (3^2-3-1)^2-(3^2-3-1)-1 = 19 is also prime. So, 3 is a member of this sequence.

Cf. A000040, A237527, A002327.

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

f = lambda x: x^2-x-1
[p for p in primes(1452) if is_prime(f(f(p)))] # Peter Luschny, Mar 02 2014

A237527(n) = A165900(a(n)). - M. F. Hasler, Mar 01 2014

A236387 Numbers n such that sigma(n) is an oblong number.

Original entry on oeis.org

5, 6, 11, 19, 20, 26, 28, 29, 30, 39, 40, 41, 46, 51, 55, 58, 71, 86, 89, 99, 104, 109, 114, 116, 117, 125, 131, 135, 158, 177, 181, 201, 202, 203, 209, 216, 226, 236, 239, 245, 271, 278, 306, 336, 340, 352, 377, 379, 398, 410, 411, 419, 428, 442, 447, 461

Offset: 1

Joseph L. Pe, Jan 24 2014

An oblong number (A002378) is of the form k(k+1) where k is a natural number.

The subsequence of prime terms is A002327 (primes of form n^2 - n - 1). - Michel Marcus, Jan 09 2015

			sigma(40) = 90 = 9*10, an oblong number; so 40 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000203, A002327, A002378.

Select[Range[500], IntegerQ@ Sqrt[1+4*DivisorSigma[1, #]] &] (* Giovanni Resta, Jan 24 2014 *)

a(12)-a(56) from Giovanni Resta, Jan 24 2014

cp's OEIS Frontend

A088502 Numbers n such that (n^2 - 5)/4 is prime.

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A078179 a(n) is the smallest prime of the form n^k + n - 1 with k >= 2.

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A126435 Primes of the form n^7-n-1.

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A126437 Primes of the form k^8-k-1.

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A140869 Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1.

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Magma

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A237527 Numbers n of the form p^2-p-1 = A165900(p), for prime p, such that n^2-n-1 = A165900(n) is also prime.

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A126438 Primes of the form n^9-n-1.

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Mathematica

A171771 Primes of form n^6-(n+1)^5.

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