A002328 -id:A002328 - cp's OEIS frontend

A173180 Numbers k such that k^5-k^4-k^3-k^2-k-1 is prime.

4, 6, 8, 14, 18, 20, 24, 26, 28, 32, 40, 42, 50, 58, 62, 68, 72, 100, 104, 120, 122, 140, 150, 174, 184, 192, 210, 234, 240, 260, 266, 278, 288, 300, 306, 326, 346, 366, 404, 432, 444, 460, 464, 466, 470, 484, 488, 512, 516, 526, 538, 556, 562, 564, 570, 584

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

nonn

All terms are even. - Robert Israel, Apr 11 2019

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

Cf. A002328, A008864, A162294, A173179

filter:= k -> isprime( k^5-k^4-k^3-k^2-k-1):
select(filter, 2*[$1..500]); # Robert Israel, Apr 11 2019

f[n_]:=n^5-n^4-n^3-n^2-n-1;Select[Range[7! ],PrimeQ[f[ #1]]&]
Select[Range[2,600,2],PrimeQ[#^5-Total[#^Range[0,4]]]&] (* Harvey P. Dale, Sep 26 2023 *)

{k: A125083(k) in A000040}. [R. J. Mathar, Feb 13 2010]

A182438 Numbers n such that neither n^2+n-1 nor n^2-n-1 is prime.

Original entry on oeis.org

1, 18, 23, 33, 34, 37, 43, 52, 58, 62, 63, 72, 73, 74, 75, 78, 79, 80, 81, 82, 88, 91, 92, 98, 99, 105, 106, 107, 108, 109, 110, 111, 112, 113, 117, 118, 119, 122, 123, 124, 128, 129, 133, 136, 137, 143, 147, 151, 152, 157, 162, 166, 167, 168, 173

Offset: 1

Alex Ratushnyak, Apr 28 2012

			18^2+18-1=341 is not prime, and 18^2-18-1=305 is not prime, so 18 is in the sequence.

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

Cf. A002328, A045546.

[n: n in [1..180] | not IsPrime(n^2+n-1) and not IsPrime(n^2-n-1)]; // Vincenzo Librandi, Jan 19 2013

Select[Range[500], !PrimeQ[#^2 + # - 1] && !PrimeQ[#^2 - # - 1] &] (* Vincenzo Librandi, Jan 19 2013 *)
Select[Range[200],NoneTrue[#^2+{#-1,-#-1},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 04 2018 *)

is(n)=!isprime(n^2+n-1) && !isprime(n^2-n-1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) ~ n. - Charles R Greathouse IV, Jun 13 2017

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

A291689 Numbers n such that n^2 +- n +- 1 are all composite.

Original entry on oeis.org

23, 37, 43, 52, 73, 74, 82, 88, 92, 98, 107, 108, 109, 113, 122, 123, 124, 128, 129, 133, 136, 137, 152, 157, 166, 178, 179, 183, 198, 201, 202, 205, 208, 211, 212, 213, 214, 217, 222, 223, 224, 227, 228, 229, 235, 238, 239, 243, 250, 251, 252, 253, 254, 255, 256, 257, 261, 262, 270, 271, 274

Offset: 1

Robert Israel, Aug 29 2017

nonn

Numbers n such that A291654(n)=1.

Complement of union of A002328, A002384, A045546 and A055494.

			a(1)=23 is in the sequence because 23^2 - 23 - 1 = 505, 23^2 - 23 + 1 = 507, 23^2 + 23 - 1 = 551, 23^2 + 23 + 1 = 553 are all composite.

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

Cf. A002328, A002384, A045546, A055494, A291654.

select(t -> not ormap(isprime, {t^2+t+1,t^2+t-1,t^2-t+1,t^2-t-1}), [$1..1000]);

Select[Range@ 300, Function[t, AllTrue[t^2 + Map[Total[{t, 1} #] &, Tuples[{1, -1}, 2]], ! PrimeQ@ # &]]] (* Michael De Vlieger, Aug 29 2017 *)

is(n)=my(n2=n^2); !isprime(n2+n+1) && !isprime(n2+n-1) && !isprime(n2-n+1) && !isprime(n2-n-1) \\ Charles R Greathouse IV, Aug 30 2017

a(n) ~ n. - Charles R Greathouse IV, Aug 30 2017

A328525 Numbers k such that (k-1)k(k+1) = (k-1)(1+u) = k(1+v) = (k+1)*(1+w) with primes u, v, w.

Original entry on oeis.org

3, 5, 9, 21, 55, 131, 145, 155, 231, 259, 265, 449, 495, 561, 595, 1045, 1051, 1365, 1409, 1491, 1549, 1849, 1989, 2001, 2101, 2469, 2785, 3365, 3621, 3641, 3669, 3845, 3911, 4285, 4951, 5181, 5465, 6049, 6699, 7189, 7229, 8219, 8629, 9175, 9521, 9539, 9631, 9729

Offset: 1

Frank Ellermann, Feb 24 2020

nonn

			3 is a term because 2*3*4 = 2*(1+11) = 3*(1+7) = 4*(1+5) with primes 11, 7, 5.
9 is a term because 8*9*10 = 8*(1+89) = 9*(1+79) = 10*(1+71) with primes 89, 79, 71.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040.

Intersection of A002328, A028870 and A045546.

q:= k-> andmap(isprime, (t-> [t-1, t-k, t+k])(k^2-1)):
select(q, [$1..10000])[];  # Alois P. Heinz, Feb 25 2020

Select[Range[2, 10^4], AllTrue[{(# - 1)*#, #*(# + 1), (# + 1)*(# - 1)} - 1, PrimeQ] &] (* Amiram Eldar, Feb 24 2020 *)

isok(k) = isprime(k*(k+1)-1) && isprime((k+1)*(k-1)-1) && isprime(k*(k-1)-1); \\ Michel Marcus, Feb 25 2020

S = 3
do N = 5 to 595 by 2
   if NOPRIME( N*N +N -1 ) then  iterate N
   if NOPRIME( N*N    -2 ) then  iterate N
   if NOPRIME( N*N -N -1 ) then  iterate N
   S = S || ',' N
end N
say S

More terms from Amiram Eldar, Feb 24 2020

A089362 Numbers n such that n^2 - 5n + 5 is prime.

Original entry on oeis.org

5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 22, 23, 24, 27, 29, 31, 33, 34, 38, 41, 42, 44, 47, 48, 49, 51, 53, 56, 57, 58, 59, 62, 63, 67, 68, 69, 71, 73, 79, 86, 88, 89, 92, 96, 97, 99, 103, 104, 106, 117, 118, 123, 128, 129, 133, 134, 137, 141, 143, 144, 147, 148, 151, 152, 156, 157, 158, 161, 162, 163, 166, 167, 172

Offset: 1

Giovanni Teofilatto, Dec 27 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

Cf. A002328.

is(n)=isprime(n^2-5*n+5) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A002328(n)+2.

Corrected and extended by Georg Fischer, Jun 20 2020

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

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

A173180 Numbers k such that k^5-k^4-k^3-k^2-k-1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A182438 Numbers n such that neither n^2+n-1 nor n^2-n-1 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A291689 Numbers n such that n^2 +- n +- 1 are all composite.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A328525 Numbers k such that (k-1)*k*(k+1) = (k-1)*(1+u) = k*(1+v) = (k+1)*(1+w) with primes u, v, w.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Rexx

Extensions

A089362 Numbers n such that n^2 - 5n + 5 is prime.

Views

Author

Keywords

References

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

A328525 Numbers k such that (k-1)k(k+1) = (k-1)(1+u) = k(1+v) = (k+1)*(1+w) with primes u, v, w.