A002327 -id:A002327 - cp's OEIS frontend

A241670 Semiprimes of the form n^4 - n^3 - n - 1.

187, 1073, 8989, 35657, 61423, 151979, 1632923, 2495959, 8345537, 9658823, 18687173, 49194347, 64880909, 77244217, 179502923, 250046873, 451259573, 502874849, 588444323, 651263839, 830296829, 1723401587, 1935548789, 4552183739, 4839132407, 8739047573, 13324055659

Offset: 1

K. D. Bajpai, Aug 09 2014

Since n^4 - n^3 - n - 1 = (n^2 + 1)*(n^2 - n - 1), it is a must that (n^2 + 1) and (n^2 - n - 1) both should be prime.
Primes of the form (n^2+1) are at A002496.
Primes of the form (n^2-n-1) are at A002327.

			187 is in the sequence because 4^4 - 4^3 - 4 - 1 = 187 = 11 * 17, which is semiprime.
1073 is in the sequence because 6^4 - 6^3 - 6 - 1 = 1073 = 29 * 37, which is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..5330

Cf. A000040, A001358, A002327, A002496.

IsSemiprime:= func; [s: n in [1..400] | IsSemiprime(s) where s is n^4 - n^3 - n - 1]; // Vincenzo Librandi, Aug 10 2014

select(k -> numtheory:-bigomega(k)=2, [seq(x^4 - x^3 - x - 1, x=1..1000)]);

Select[Table[n^4 - n^3 - n - 1, {n, 500}], PrimeOmega[#] == 2 &]

for(n=1,10^4,p=n^2+1;q=n^2-n-1;if(isprime(p)&&isprime(q),print1(p*q,", "))) \\ Derek Orr, Aug 09 2014

A245042 Primes of the form (k^2+4)/5.

Original entry on oeis.org

17, 73, 89, 193, 337, 521, 953, 1009, 1249, 1657, 2377, 2833, 3329, 3433, 4441, 4561, 5849, 6553, 7297, 8081, 8737, 9769, 11617, 12401, 12601, 13417, 15569, 16937, 17881, 18121, 20353, 21649, 27529, 28729, 29033, 30577, 33457, 35449, 36809, 46273, 49801

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

Also equal to primes p such that 5*p-4 is a perfect square.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A154418.

Cf. A154616, A002327, A066436.

Select[(Range[500]^2+4)/5,PrimeQ] (* Harvey P. Dale, Jul 13 2014 *)

import sympy
L = (k**2 + 4 for k in range(10**3))
[n//5 for n in L if n % 5 == 0 and sympy.ntheory.isprime(n//5)]

A245045 Primes of the form (k^2+2)/6.

Original entry on oeis.org

3, 11, 17, 43, 67, 113, 131, 193, 241, 353, 523, 641, 683, 1291, 1601, 1667, 1873, 2017, 2243, 2731, 3083, 3361, 3851, 4483, 4817, 4931, 5281, 5521, 7211, 8363, 8513, 8971, 9283, 9923, 10753, 11971, 13633, 16433, 17713, 18371, 18593, 19267, 21841, 22571

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

			When k=4, (k^2+2)/6 = 3 is prime, so 4 is a member of the sequence. since putting k = 0, 1, 2, or 3 does not give a prime, so 4 is the first term.

Chai Wah Wu, Table of n, a(n) for n = 1..1500

Cf. A154616, A002327, A066436. First 5 terms equal to A078116. First 4 terms equal to A127996.

import sympy
[(k**2+2)/6 for k in range(10**6) if sympy.ntheory.isprime((k**2+2)/6) & ((k**2+2)/6).is_integer()]

A265006 Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

Original entry on oeis.org

5, 7, 11, 13, 29, 31, 41, 43, 71, 73, 239, 241, 419, 421, 461, 463, 599, 601, 1481, 1483, 1721, 1723, 2549, 2551, 2969, 2971, 3539, 3541, 4421, 4423, 8009, 8011, 10301, 10303, 17291, 17293, 19181, 19183, 20021, 20023, 23561, 23563, 24179, 24181, 27059, 27061, 31151, 31153, 35531, 35533

Offset: 1

Bill McEachen, Nov 29 2015

This is a subset of A002327 and A002383 taken together. Note that 3 is not a member, as the pairing (3, 5) is excluded as defined, as 3 and 5 associate to different centers.
The corresponding n are in A088485.
The average of each twin prime pair is an oblong number (A002378). - Michel Marcus, Feb 04 2017

			For k = 6, k^2 + k = 6^2 + 6 = 42, and (41,43) is a twin prime pair, so 41 and 43 are in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3594 from G. C. Greubel)

Cf. A002327, A002378, A002383, A088485, A088486, A306889.

&cat[[n^2+n-1, n^2+n+1]: n in [0..250]| IsPrime(n^2+n-1) and IsPrime(n^2+n+1)]; // Vincenzo Librandi, Feb 05 2017

{#^2 + # - 1, #^2 + # + 1} & /@ Select[Range@ 200, PrimeQ[#^2 + # - 1] && PrimeQ[#^2 + # + 1] &] // Flatten (* Michael De Vlieger, Nov 30 2015 *)
Flatten[Select[Table[n^2 + n + {-1, 1}, {n, 0, 200}], And@@PrimeQ[#] &]] (* Vincenzo Librandi, Feb 05 2017 *)

PARI
```
genit()={my(maxx=1000);n=0;while(n
				
```

a(2n-1) = A088486(n). a(2n)=2+a(2n-1).

A126439 Least prime of the form x^n-x-1.

Original entry on oeis.org

5, 5, 13, 29, 61, 2097143, 1679609, 509, 1021, 8589934583, 4093, 67108859, 16381, 470184984569, 4294967291, 2218611106740436979, 68719476731, 1350851717672992079, 1048573, 10460353199, 4194301, 20013311644049280264138724244295359, 16777213, 108347059433883722041830239, 20282409603651670423947251285999, 58149737003040059690390159, 72057594037927931, 536870909, 999999999999999999999999999989

Offset: 2

Artur Jasinski, Dec 26 2006, Jan 19 2007

nonn

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434, A126435, A126436, A126437, A126438.

a = {}; Do[k = 2; While[ ! PrimeQ[k^n -k - 1], k++ ]; AppendTo[a, k^n - k - 1], {n, 2, 30}]; a (*Artur Jasinski*)

A241975 Numbers n such that n^4 - n^3 - n - 1 is a semiprime.

Original entry on oeis.org

4, 6, 10, 14, 16, 20, 36, 40, 54, 56, 66, 84, 90, 94, 116, 126, 146, 150, 156, 160, 170, 204, 210, 260, 264, 306, 340, 350, 386, 396, 406, 420, 464, 474, 496, 570, 634, 674, 696, 700, 716, 740, 764, 780, 816, 826, 864, 890, 966, 1054, 1070, 1094, 1106, 1144

Offset: 1

Vincenzo Librandi, Aug 10 2014

Since n^4 - n^3 - n - 1 = (n^2 + 1)*(n^2 - n - 1), these are also numbers n such that n^2 + 1 and n^2 - n - 1 are both prime. Numbers in the intersection of A005574 and A002328. - Derek Orr, Aug 10 2014 [Sequence numbers corrected by Jens Kruse Andersen, Aug 11 2014]

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Cf. A001358, A002327, A002496, A085722, A241670, A243436.

IsSemiprime:=func; [ n: n in [2..1500] | IsSemiprime(n^4 - n^3 - n - 1)];

Select[Range[2000], PrimeOmega[#^4 - #^3 - # - 1]==2 &]

for(n=1,10^4,if(isprime(n^2+1)&&isprime(n^2-n-1),print1(n,", "))) \\ Derek Orr, Aug 10 2014

A242930 Primes of the form (k^2+7)/11.

Original entry on oeis.org

37, 53, 193, 373, 421, 673, 1061, 2213, 2753, 3637, 4481, 5237, 5413, 7333, 7541, 8513, 8737, 9781, 11393, 12853, 14401, 15733, 17761, 19237, 21121, 25153, 25537, 27701, 29537, 34273, 34721, 39841, 42533, 47653, 50593, 51137

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

Also equal to primes p such that 11*p-7 is a perfect square.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A154616, A002327, A066436.

import sympy
[(k**2+7)/11 for k in range(10**6) if sympy.ntheory.isprime((k**2+7)/11) & ((k**2+7)/11).is_integer()]

A172192 Numbers k such that k^6 - (k+1)^5 is prime.

Original entry on oeis.org

4, 9, 10, 12, 14, 19, 29, 46, 57, 59, 66, 71, 72, 84, 85, 90, 95, 96, 97, 114, 119, 122, 155, 157, 190, 191, 204, 207, 212, 221, 222, 244, 251, 256, 276, 285, 286, 289, 294, 300, 301, 307, 319, 320, 337, 344, 355, 359, 380, 382, 392, 400, 411, 422, 426, 441, 451

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jan 29 2010

Corresponding primes are in A171771. Negative values of primes are obtained for 1 and 2.

			4^6-(4+1)^5 = 971 is prime, so 4 is in the sequence.
5^6-(5+1)^5 = 7849 = 47*167 is composite, so 5 is not in the sequence.
9^6-(9+1)^5 = 431441 is prime, so 9 is in the sequence.

Cf. A171771, A002327 (primes of form n^2-n-1), A140719 (primes of form n^3-(n+1)^2), A087191 (primes of form n^4-(n+1)^3).

[ n: n in [1..460] | IsPrime(p) and p gt 0 where p is n^6-(n+1)^5 ];

Select[Range[3,500],PrimeQ[#^6-(#+1)^5]&] (* Harvey P. Dale, Apr 25 2011 *)

is(n)=isprime(n^6-(n+1)^5) \\ Charles R Greathouse IV, Jun 13 2017

Edited, extended, non-specific references removed and MAGMA program added by Associate Editors OEIS, Mar 05 2010

A235484 Square numbers n such that n^2 - n - 1 is prime.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 121, 196, 289, 324, 361, 529, 625, 729, 1024, 1296, 1681, 1849, 2916, 3600, 4225, 4761, 5184, 5929, 6400, 6724, 6889, 7569, 7744, 8464, 8649, 9604, 12100, 13689, 14641, 14884, 15876, 16129, 18225, 18496, 19044, 22201, 22500, 24025, 24649, 25281, 28224

Offset: 1

Zak Seidov, Apr 13 2014

nonn

Or, squares in A002328: a(1) = 4 = A002328(2), a(2) = 9 = A002328(6), a(1) = 16 = A002328(11).

The corresponding primes, 11, 71, 239, 599, 1259, 2351, 14519, 38219, 83231, 104651, 129959, 279311, 389999, 530711, 1047551, 1678319, are a subsequence of A002327.

Intersection of A002328 and A000290. Cf. A002327.

Select[Table[n^2, {n, 100}], PrimeQ[#^2 - # - 1] &]

list(lim)=my(v=List()); for(n=2,sqrtint(lim\1), if(isprime(n^2-n-1), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 13 2017

cp's OEIS Frontend

A241670 Semiprimes of the form n^4 - n^3 - n - 1.

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A245042 Primes of the form (k^2+4)/5.

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A245045 Primes of the form (k^2+2)/6.

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A265006 Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

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A126439 Least prime of the form x^n-x-1.

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A241975 Numbers n such that n^4 - n^3 - n - 1 is a semiprime.

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Magma

Mathematica

PARI

A242930 Primes of the form (k^2+7)/11.

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Python

A172192 Numbers k such that k^6 - (k+1)^5 is prime.

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