A121326 -id:A121326 - cp's OEIS frontend

A248887 Primes p of the form 4m^2+1 such that q=4p^2+1 and r=4q^2+1 are prime.

677, 6635777, 28132417, 156400037, 234518597, 381655297, 386751557, 403849217, 820020497, 1215498497, 1298449157, 1539463697, 1580539537, 1839037457, 2072616677, 2774550277, 2969814017, 6147500837, 6194319617, 6703351877, 6937890437

Offset: 1

Zak Seidov, Mar 05 2015

nonn

All terms == 7 mod 10. Subsequence of A121834.

Cf. A001912, A121326.

A271725 T(n,k) is an array read by rows, with n > 0 and k=1..4, where row n gives four prime numbers in increasing order with locations in right angles of each concentric square drawn on a distorted version of the Ulam spiral.

Original entry on oeis.org

3, 7, 17, 19, 13, 23, 37, 41, 307, 359, 401, 419, 13807, 14159, 14401, 14519, 41413, 42023, 42437, 42641, 6317683, 6325223, 6330257, 6332771, 22958473, 22972847, 22982437, 22987229, 39081253, 39100007, 39112517, 39118769, 110617807, 110649359, 110670401, 110680919

Offset: 1

Michel Lagneau, Apr 13 2016

nonn

See the illustration for more information.
Conjecture: there is an infinity of concentric squares having a prime number in each right angle. The number 5 is the center of all the squares.
It seems that the drawing of an infinite number of concentric squares having a prime number in each corner is impossible in an Ulam spiral. But with a slight distortion of this space, the problem becomes possible.
The illustration (see the link) shows the new version of a spiral with two remarkable orthogonal diagonals containing four classes of prime numbers given by the sequences A125202, A121326, A028871 and A073337 supported by four line segments. These intersect at a single point represented by the prime number 5.
The sequence of the corresponding length of the sides is {s(k)} = {2, 4, 18, 118, 204, 2514, 4792, 6252, 10518, 14032, 16752, 17598, ...}
The primes are defined by the polynomials: [4*m^2-10*m+7, (2*m-1)^2-2, 4*m^2+1, 4*(m+1)^2-6*(m+1)+1]. The sequence of the corresponding m is {b(k)} = {2, 3, 10, 60, 103, 1258, 2397, 3127, 5260, 7017, 8377, 8800, 10375, 11518, 11523, 12498, 15415, 15888, ...} with the relation b(k) = 1 + s(k)/2.
The array begins:
      3,     7,    17,    19;
     13,    23,    37,    41;
    307,   359,   401,   419;
  13807, 14159, 14401, 14519;
  41413, 42023, 42437, 42641;
  ...
Construction of the spiral (see the illustration in the link):
                .  .  .  .  .  .  .  .  .  .  .  .
               .  42  41  40  39  38  37   .  .  .
                                       |
               .  43  20  19  18  17  36  35  .  .
                                   |
               .  .   21   6   5  16  15  34  .  .
                               |
               .  .   22   7   4   3  14  33  .  .
               .  .   23   8   1   2  13  32  .  .
               .  .   24   9  10  11  12  31  .  .
               .  .   25  26  27  28  29  30  .  .
                   .  .  .  .  .  .  .  .  .  .  .
The first squares of center 5 having a prime number in each vertex are:
    19  18  17      41  40  39  38  37
     6   5  16      20  19  18  17  36
     7   4  3       21   6   5  16  15   . . . .
                    22   7   4   3  14
                    23   8   1   2  13

Michel Lagneau, Illustration

Cf. A028871, A073337, A121326, A125202, A200975.

for n from 1 to 10000 do :
  x1:=4*n^2-10*n+7:x2:=(2*n-1)^2-2:
  x3:=4*(n+1)^2-6*(n+1)+1:x4:=4*n^2+1:
   if isprime(x1) and isprime(x2) and isprime(x3) and isprime(x4)
    then
     printf("%d %d %d %d %d \n",n,x1,x2,x4,x3):
    else
    fi:
od:

A296422 Primes that can be represented in the form b^n+1 or b^n-1 where b >= 2 and n >= 2.

Original entry on oeis.org

3, 5, 7, 17, 31, 37, 101, 127, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8191, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177, 52901

Offset: 1

Nathaniel J. Strout, Dec 12 2017

nonn

Union of A000668 and A121326. - Andrey Zabolotskiy, Dec 21 2017

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

Cf. A000040 (primes), A001597 (perfect powers).

Cf. A000668 (Mersenne primes), A121326.

N:= 10^5: # to get terms <= N
R:= 3:
for b from 2 while b^2+1 <= N do
  p:= 2:
  do
    p:= nextprime(p);
    if b^p-1 > N then break fi;
    if isprime(b^p-1) then R:= R, b^p-1 fi;
  od:
  p:= 1:
  do
    p:= 2*p;
    if b^p+1 > N then break fi;
    if isprime(b^p+1) then R:= R, b^p+1 fi;
  od;
od:
sort(convert({R},list)); # Robert Israel, Jan 08 2018

Select[Prime@ Range[2, 10^4], AnyTrue[# + {-1, 1}, Or[# == 1, GCD @@ FactorInteger[#][[All, -1]] > 1] &] &] (* Michael De Vlieger, Dec 13 2017 *)

lista(nn) = {forprime(p=2, nn, if ((p==2) || ispower(p+1) || ispower(p-1), print1(p, ", ")); ); } \\ Michel Marcus, Dec 13 2017

A144852 a(n) = number of distinct prime divisors (taken together) of numbers of the form 4x^2+1 for x<=10^n.

Original entry on oeis.org

9, 87, 836, 8000, 78124, 766585, 7556731, 74771106, 741554656, 7366252759, 73261462211, 729280694469

Offset: 1

Artur Jasinski & Bernhard Helmes (bhelmes(AT)gmx.de), Sep 22 2008

nonn

Primes of the form 4x^2+1 see A121326(n) = A002496(n+1).

Bernhard Helmes, Prime sieving on the polynomial f(n)=4n^2+1.

Cf. A002383, A121326, A143835, A143868, A144848, A144850, A144851.

d = 10; l = 0; p = 4; c = {}; a = {}; Do[k = p x^2 + 1; b = Divisors[k]; Do[If[PrimeQ[b[[n]]], AppendTo[a, b[[n]]]], {n, 1, Length[b]}]; If[x == d, a = Union[a]; l = Length[a]; d = 10 d; Print[l]; AppendTo[c, l]], {x, 1, 10000}]; c (*Artur Jasinski*)

Fixed broken link, corrected and extended to agree with website. - Ray Chandler, Jun 30 2015

A248892 Primes p of the form 4m^2+1 such that q=4p^2+1, r=4q^2+1 and s=4r^2+1 are all prime.

Original entry on oeis.org

1565891838737, 1985917919077, 2060476510097, 5590084720897, 39623323626437, 94860314619877, 114027286862737, 115071875848337, 117140013119377, 136739205150917, 246382184192357, 254109295929637, 265883157493777, 340055949647237, 378534223873937

Offset: 1

Zak Seidov, Mar 05 2015

nonn

Corresponding values of k: 625678,704613,717718,1182168,3147353,4869813,5339178,5363578,5411562,5846777,7848283,7970403,8152962,9220303,9727978.

Subsequence of A248887. Cf. A001912, A121326, A121834, A248887.

apQ[p_]:=Module[{q=4p^2+1,r},r=4q^2+1;AllTrue[{p,q,r,4r^2+1},PrimeQ]]; Select[ 4*Range[ 10^7]^2+1,apQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 28 2019 *)

cp's OEIS Frontend

A248887 Primes p of the form 4m^2+1 such that q=4p^2+1 and r=4q^2+1 are prime.

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A271725 T(n,k) is an array read by rows, with n > 0 and k=1..4, where row n gives four prime numbers in increasing order with locations in right angles of each concentric square drawn on a distorted version of the Ulam spiral.

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Maple

A296422 Primes that can be represented in the form b^n+1 or b^n-1 where b >= 2 and n >= 2.

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Maple

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PARI

A144852 a(n) = number of distinct prime divisors (taken together) of numbers of the form 4x^2+1 for x<=10^n.

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A248892 Primes p of the form 4m^2+1 such that q=4p^2+1, r=4q^2+1 and s=4r^2+1 are all prime.

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Mathematica