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

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

Author

Michel Lagneau, Mar 14 2016

Keywords

• nonn

Comments

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.

Links

• Michel Lagneau, Illustration

Crossrefs

Cf. A087277, A172979.

Programs

Maple

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: