A087277 -id:A087277 - cp's OEIS frontend

A231612 Numbers n such that the four fourth-degree cyclotomic polynomials are simultaneously prime.

2, 90750, 194468, 229592, 388332, 868592, 1054868, 1148390, 1380380, 1415920, 1461372, 1496010, 1614800, 1706398, 1992210, 2439042, 2478212, 2644498, 2791910, 3073300, 3264448, 3824370, 3892780, 3939222, 3941938, 4425970, 4468980, 4594138, 4683700

Offset: 1

T. D. Noe, Dec 11 2013

nonn

The polynomials are cyclotomic(5,x) = 1 + x + x^2 + x^3 + x^4, cyclotomic(8,x) = 1 + x^4, cyclotomic(10,x) = 1 - x + x^2 - x^3 + x^4, and cyclotomic(12,x) = 1 - x^2 + x^4. The numbers 5, 8, 10, and 12 are in the fourth row of A032447.

By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).
Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

Select[Range[5000000], PrimeQ[Cyclotomic[5, #]] && PrimeQ[Cyclotomic[8, #]] && PrimeQ[Cyclotomic[10, #]] && PrimeQ[Cyclotomic[12, #]] &]
Select[Range[47*10^5],AllTrue[Thread[Cyclotomic[{5,8,10,12},#]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 22 2018 *)

A231613 Numbers n such that the four sixth-degree cyclotomic polynomials are simultaneously prime.

Original entry on oeis.org

32034, 162006, 339105, 458811, 1780425, 2989119, 2993100, 3080205, 4375404, 6129597, 6280221, 7565142, 8489820, 10268277, 11343741, 12065076, 13067295, 13333182, 15866508, 16472802, 17040537, 18028605, 19066758, 22633629, 24256362, 24365259, 25031349

Offset: 1

T. D. Noe, Dec 11 2013

nonn

The polynomials are cyclotomic(7,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6, cyclotomic(9,x) = 1 + x^3 + x^6, cyclotomic(14,x) = 1 - x + x^2 - x^3 + x^4 - x^5 + x^6, and cyclotomic(18,x) = 1 - x^3 + x^6. The numbers 7, 9, 14 and 18 are in the sixth row of A032447.

By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

t = {}; n = 0; While[Length[t] < 30, n++; If[PrimeQ[Cyclotomic[7, n]] && PrimeQ[Cyclotomic[9, n]] && PrimeQ[Cyclotomic[14, n]] && PrimeQ[Cyclotomic[18, n]], AppendTo[t, n]]]; t
Select[Range[251*10^5],AllTrue[Cyclotomic[{7,9,14,18},#],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 29 2016 *)

A231614 Numbers n such that the five eighth-degree cyclotomic polynomials are simultaneously prime.

Original entry on oeis.org

4069124, 8919014, 8942756, 46503870, 75151624, 82805744, 189326670, 197155324, 271490544, 365746304, 648120564, 1031944990

Offset: 1

T. D. Noe, Dec 11 2013

The polynomials are cyclotomic(15,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8, cyclotomic(16,x) = 1 + x^8, cyclotomic(20,x) = 1 - x^2 + x^4 - x^6 + x^8, cyclotomic(24,x) = 1 - x^4 + x^8, and cyclotomic(30,x) = 1 + x - x^3 - x^4 - x^5 + x^7 + x^8. The numbers 15, 16, 20, 24 and 30 are in the eighth row of A032447.

By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).

t = {}; n = 0; While[Length[t] < 6, n++; If[PrimeQ[Cyclotomic[15, n]] && PrimeQ[Cyclotomic[16, n]] && PrimeQ[Cyclotomic[20, n]] && PrimeQ[Cyclotomic[24, n]] && PrimeQ[Cyclotomic[30, n]], AppendTo[t, n]]]; t

Extended to 12 terms by T. D. Noe, Dec 13 2013

A233512 The first n cyclotomic polynomials are simultaneously prime for these arguments.

Original entry on oeis.org

3, 4, 6, 150, 1068630, 6770610

Offset: 1

T. D. Noe, Dec 13 2013

The first six cyclotomic polynomials are x - 1, x + 1, x^2 + x + 1, x^2 + 1, x^4 + x^3 + x^2 + x + 1, and x^2 - x + 1.
By Schinzel's hypothesis H, this sequence is defined for all n.
a(7) > 2*10^9.

			At x = 3, x-1 = 2, which is prime. At x = 4, x-1 = 3 and x+1 = 5, which are both prime. At x = 6, x-1 = 5, x+1 = 7, and x^2+x+1 = 43, which are all prime.

See A087277.

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).
Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

t = {}; n = 0; len = 0; While[len < 6, n++; found = True; i = 1; While[found && i <= len + 1, found = PrimeQ[Cyclotomic[i, n]]; i++]; If[found && i > len + 1, AppendTo[t, n]; len++]]; t

A270269 Prime numbers with locations of right angle turns in the Ulam square spiral that are vertices of isosceles right triangles.

Original entry on oeis.org

3, 5, 7, 31, 37, 43, 8011, 8101, 8191, 920641, 921601, 922561, 3894703, 3896677, 3898651, 5902471, 5904901, 5907331, 7450171, 7452901, 7455631, 7482961, 7485697, 7488433, 36066031, 36072037, 36078043, 37155121, 37161217, 37167313, 39759331, 39765637, 39771943

Offset: 1

Michel Lagneau, Mar 14 2016

nonn

See the illustration for more information.
Subsequence of A172979. This sequence is probably infinite.
An interesting property: the sequence of the differences between prime numbers that are vertices for each triangle is the sequence {2, 6, 90, 960, 1974, 2430, 2730, 2736, 6006, 6096, 6306, ...} = A087277: numbers n such that the three second-degree cyclotomic polynomials x^2 + 1, x^2 - x + 1 and x^2 + x + 1 are simultaneously prime.
For example:
2 = 5 - 3 = 7 - 5;
6 = 37 - 31 = 43 - 37;
90 = 8101 - 8011 = 8191 - 8101.
Consequence: a(3n) + A087277(n) is a square. The corresponding sequence of the squares is {3^2, 7^2, 91^2, 961^2, 1975^2, 2431^2, 2731^2, 2737^2, 6007^2, ...}.
Examples:
a(3) + A087277(1) = 7 + 2 = 3^2;
a(6) + A087277(2) = 43 + 6 = 7^2;
a(9) + A087277(3) = 8191 + 90 = 91^2.

Michel Lagneau, Illustration

Cf. A087277, A172979.

nn:=20000:T:=array(1..nn):a0:=1:kk:=0:
for p from 1 to nn do :
   a1:=a0+floor(p/2):a0:=a1:
    if  isprime(a1)
     then
     kk:=kk+1:T[kk]:=a1:
     else
    fi:
  od:
   for n from 1 to kk-2 do:
    d1:=T[n+2]-T[n+1]:d2:=T[n+1]-T[n]:
     if d1=d2
     then
      printf("%d %d %d \n", T[n], T[n+1], T[n+2]):
      else
     fi:
   od:

cp's OEIS Frontend

A231612 Numbers n such that the four fourth-degree cyclotomic polynomials are simultaneously prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

A231613 Numbers n such that the four sixth-degree cyclotomic polynomials are simultaneously prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

A231614 Numbers n such that the five eighth-degree cyclotomic polynomials are simultaneously prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Extensions

A233512 The first n cyclotomic polynomials are simultaneously prime for these arguments.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A270269 Prime numbers with locations of right angle turns in the Ulam square spiral that are vertices of isosceles right triangles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple