A091567 -id:A091567 - cp's OEIS frontend

A119609 p^2-p-1 that is not prime, where p is prime.

1, 155, 341, 505, 1331, 1639, 1805, 2755, 3421, 5255, 6161, 6805, 7831, 10505, 11341, 11771, 12655, 18631, 22649, 24491, 26405, 27721, 29755, 31861, 36289, 37055, 39401, 44309, 49505, 51301, 52211, 54055, 56881, 62749, 65791, 68905, 73169

Offset: 1

Alexander Adamchuk, Jul 27 2006

nonn

All prime factors of a(n) {5,11,19,29,31,41,59,61,..} belong to A038872 Primes congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square mod p. A091568 Primes of the form p^2-p-1, where p is prime. A091567 Primes p such that p^2-p-1 is prime.

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

Cf. A091568, A091567, A038872.

[q: p in PrimesUpTo(300) | IsPrime(p) and not IsPrime(q) where q is p^2 - p - 1] // Vincenzo Librandi, Sep 08 2012

lst = {}; Do[If[PrimeQ[p] && ! PrimeQ[p^2 - p - 1], AppendTo[lst, p^2 - p -1]], {p, 300}]; lst (* Vincenzo Librandi, Sep 08 2012 *)

A137460 Prime numbers k such that k^2 +- (k+1) are primes.

Original entry on oeis.org

3, 5, 17, 71, 101, 131, 677, 839, 857, 1091, 1217, 2129, 2309, 2339, 2957, 3137, 3449, 3989, 4409, 6569, 6719, 6761, 7229, 8501, 8627, 8807, 9521, 9689, 9749, 10589, 10631, 11621, 11777, 11927, 12641, 13487, 13931, 14519, 15527, 15797, 16007

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2008

			3^2 +- 4 -> (5,13) primes,
5^2 +- 6 -> (19,31) primes.

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

Cf. A053182, A091567.

Select[Prime[Range[800]], PrimeQ[ #^2 - (# + 1)] && PrimeQ[ #^2 + (# + 1)] &]

A053182 INTERSECT A091567. - R. J. Mathar, Apr 19 2009

More terms from Karl Hovekamp, Jan 24 2009

A236173 Primes p such that p^2 - p - 1, p^3 - p - 1 and p^4 - p - 1 are all prime.

Original entry on oeis.org

11, 71, 11621, 28151, 32089, 37501, 39209, 45329, 66161, 76649, 114599, 122131, 136949, 154991, 202999, 228901, 243391, 270269, 296911, 313909, 318679, 333701, 343309, 359291, 369979, 371281, 371981, 373171, 373459

Offset: 1

Derek Orr, Jan 19 2014

nonn

Primes in A236171. All primes appear to end in a 1 or a 9 (congruent to either 1 mod 10 or 9 mod 10).

			228901 is prime, 228901^2 - 228901 - 1 is prime, 228901^3 - 228901 - 1 is prime, and 228901^4 - 228901 - 1 is prime. So 228901 is a member of this sequence.

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

Cf. A091567, A236168, A236071.

Select[Prime[Range[32000]],AllTrue[#^{2,3,4}-#-1,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 08 2019 *)

s=[]; forprime(p=2, 400000, if(isprime(p^2-p-1) && isprime(p^3-p-1) && isprime(p^4-p-1), s=concat(s, p))); s \\ Colin Barker, Jan 20 2014

import sympy
from sympy import isprime
{print(p) for p in range(10**6) if isprime(p) and isprime(p**2-p-1) and isprime(p**3-p-1) and isprime(p**4-p-1)}

A238447 Primes p such that f(p) and f(f(p)) are both prime, where f(x) = x^2-x-1.

Original entry on oeis.org

3, 487, 617, 677, 751, 1201, 1289, 1579, 1747, 2027, 2267, 2269, 2309, 3259, 3947, 4457, 4567, 4621, 4637, 4799, 4951, 5701, 6029, 6991, 7151, 7687, 7867, 9187, 9209, 9341, 9587, 9829, 11321, 12301, 12541, 12781, 13177, 13649, 15919, 16349

Offset: 1

Derek Orr, Feb 26 2014

nonn

Intersection of A230026 and A091567.

Note that f(f(f(p))) is always composite. - Zak Seidov, Nov 10 2014

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

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A237642, A230026, A091567.

Select[Prime[Range[2000]],AllTrue[Rest[NestList[#^2-#-1&,#,2]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 03 2017 *)

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

A243544 Primes p such that p^2 - p + 1 is semiprime.

Original entry on oeis.org

5, 11, 29, 37, 41, 43, 53, 61, 71, 73, 83, 97, 109, 113, 127, 137, 149, 157, 167, 181, 191, 211, 223, 229, 241, 271, 277, 281, 307, 317, 331, 359, 389, 421, 433, 443, 461, 463, 487, 499, 547, 557, 571, 587, 601, 617, 631, 659, 661, 683, 691, 701, 709, 733, 757

Offset: 1

K. D. Bajpai, Jun 06 2014

nonn

Intersection of A000040 and A180748.

			11 is in the sequence because 11 is prime and 11^2 - 11 + 1 = 111 = 3 * 37 is semiprime.
29 is in the sequence because 29 is prime and 29^2 - 29 + 1 = 813 = 3 * 271 is semiprime.
17 is not in the sequence though 17 is prime, because 17^2 - 17 + 1 = 273 = 3 * 7 * 13, has more than two prime factors.

K. D. Bajpai, Table of n, a(n) for n = 1..7090

Cf. A000040, A001358, A053182, A053184, A065508, A091567, A180748.

with(numtheory): A243544 := proc() local a; a:=ithprime(n);  if bigomega(a^2-a+1)=2 then RETURN (a); fi; end: seq(A243544 (), n=1..200);

c = 0; Do[k = Prime[n]; If[PrimeOmega[k^2 - k + 1] == 2, c++; Print[c, " ", k]], {n, 1, 30000}];
Select[Prime[Range[150]],PrimeOmega[#^2-#+1]==2&] (* Harvey P. Dale, Oct 22 2024 *)

s=[]; forprime(p=2, 800, if(bigomega(p^2-p+1)==2, s=concat(s, p))); s \\ Colin Barker, Jun 06 2014

A290767 Primes p such that p^2 +/- p +/- 1 are all nonprimes.

Original entry on oeis.org

23, 37, 43, 73, 107, 109, 113, 137, 157, 179, 211, 223, 227, 229, 239, 251, 257, 271, 277, 283, 311, 313, 317, 347, 353, 367, 389, 439, 443, 467, 503, 509, 521, 523, 547, 557, 563, 577, 587, 593, 601, 631, 653, 661, 719, 733, 757, 797, 811, 821, 823, 829, 853, 859, 877, 883

Offset: 1

Ralf Steiner, Aug 10 2017

nonn

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

Cf. A000040, A053184, A053182, A065508, A091567, A236056.

select(p -> isprime(p) and not ormap(isprime, [p^2+p+1,p^2+p-1,p^2-p+1,p^2-p-1]), [2,seq(i,i=3..1000,2)]); # Robert Israel, Aug 10 2017

Select[Prime[Range[1000]], ! (PrimeQ[#^2 + # + 1] || PrimeQ[#^2 + # - 1] ||PrimeQ[#^2 - # + 1] || PrimeQ[#^2 - # - 1]) &]
Select[Prime[Range[200]],NoneTrue[{#^2+#+1,#^2+#-1,#^2-#+1,#^2-#-1},PrimeQ]&] (* Harvey P. Dale, Oct 13 2024 *)

is(n) = my(v=[n^2+n+1, n^2+n-1, n^2-n+1, n^2-n-1]); for(k=1, #v, if(ispseudoprime(v[k]), return(0))); 1
forprime(p=1, 900, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 10 2017

Intersection of the complements of A053184, A053182, A065508, and A091567 within the primes A000040.

A119570 Primes p such that p^2 - p - 1 is not prime.

Original entry on oeis.org

2, 13, 19, 23, 37, 41, 43, 53, 59, 73, 79, 83, 89, 103, 107, 109, 113, 137, 151, 157, 163, 167, 173, 179, 191, 193, 199, 211, 223, 227, 229, 233, 239, 251, 257, 263, 271, 277, 281, 283, 293, 311, 313, 317, 337, 347, 349, 353, 367, 373, 383, 389, 401, 431, 433

Offset: 1

Alexander Adamchuk, Jul 27 2006

nonn

Cf. A091567 (Primes p such that p^2-p-1 is prime),

A091568 (Primes of the form p^2-p-1, where p is prime).

[p: p in PrimesUpTo(700)| not IsPrime(p^2-p-1)] // Vincenzo Librandi, Jan 29 2011

Select[Prime[Range[250]], Not[PrimeQ[ #^2-#-1]]&]

A119964 Numerator of the n-th Artin product.

Original entry on oeis.org

1, 5, 19, 779, 84911, 2632241, 713337311, 1163866139, 587752400195, 476667196558145, 2856927907113011, 345688276760674331, 13819099649042566549, 4988694973304366524189, 10780569837310736058772429

Offset: 1

Alexander Adamchuk, Aug 03 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}].

Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A119534, A091568, A091567, A005596, A048296.

Table[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]],{n,1,20}]

a(n) = Numerator[ Product[ 1 - 1/(Prime[k]*(Prime[k]-1)), {k,1,n}]].

A119978 Denominator of the n-th Artin product.

Original entry on oeis.org

2, 12, 48, 2016, 221760, 6918912, 1881944064, 3079544832, 1558249684992, 1265298744213504, 7591792465281024, 919297051250393088, 36771882050015723520, 13282003796465679335424

Offset: 1

Alexander Adamchuk, Aug 03 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}].

Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A119534, A091568, A091567, A005596, A048296.

Table[Denominator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]],{n,1,20}]

a(n) = Denominator[ Product[ 1 - 1/(Prime[k]*(Prime[k]-1)), {k,1,n}]].

A268212 Numbers n of the form 2^k + 1 such that n^2 - n - 1 is a prime q (for k >= 0).

Original entry on oeis.org

3, 5, 9, 17, 65, 1025, 65537, 16777217, 67108865, 34359738369, 4503599627370497, 36028797018963969, 39614081257132168796771975169, 22300745198530623141535718272648361505980417

Offset: 1

Jaroslav Krizek, Jan 28 2016

nonn

Conjecture: subsequence of prime terms (3, 5, 17, 65537, ...) is not the same as A249759.
Corresponding values of numbers k are in A098855 (numbers n such that 4^n + 2^n - 1 is prime).
Corresponding values of primes q: 5, 19, 71, 271, 4159, 1049599, 4295032831, ...
4 out of 5 known Fermat primes (3, 5, 17, 65537) are terms; corresponding values of primes q: 5, 19, 271, 4295032831.

			17  = 2^4 + 1 is a term because 17^2 - 17 - 1 = 271 (prime).

Intersection of A002328 and A000051.

Cf. A019434, A091567, A098855, A249759.

[2^n + 1: n in [0..300] | IsPrime((2^n + 1)^2 - 2^n - 2)]

2^# + 1 &@ Select[Range[0, 300], PrimeQ[#^2 - # - 1 &@ (2^# + 1)] &] (* Michael De Vlieger, Jan 29 2016 *)

lista(nn) = {for (k=0, nn, n = 2^k+1; if (isprime(n^2-n-1), print1(n, ", ")););} \\ Michel Marcus, Mar 06 2016

cp's OEIS Frontend

A119609 p^2-p-1 that is not prime, where p is prime.

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A137460 Prime numbers k such that k^2 +- (k+1) are primes.

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A236173 Primes p such that p^2 - p - 1, p^3 - p - 1 and p^4 - p - 1 are all prime.

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A238447 Primes p such that f(p) and f(f(p)) are both prime, where f(x) = x^2-x-1.

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A243544 Primes p such that p^2 - p + 1 is semiprime.

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Maple

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A290767 Primes p such that p^2 +/- p +/- 1 are all nonprimes.

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A119570 Primes p such that p^2 - p - 1 is not prime.

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Magma

Mathematica

A119964 Numerator of the n-th Artin product.

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