cp's OEIS Frontend

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

%I A270269 #28 Sep 12 2017 21:31:40
%S A270269 3,5,7,31,37,43,8011,8101,8191,920641,921601,922561,3894703,3896677,
%T A270269 3898651,5902471,5904901,5907331,7450171,7452901,7455631,7482961,
%U A270269 7485697,7488433,36066031,36072037,36078043,37155121,37161217,37167313,39759331,39765637,39771943
%N A270269 Prime numbers with locations of right angle turns in the Ulam square spiral that are vertices of isosceles right triangles.
%C A270269 See the illustration for more information.
%C A270269 Subsequence of A172979. This sequence is probably infinite.
%C A270269 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.
%C A270269 For example:
%C A270269 2 = 5 - 3 = 7 - 5;
%C A270269 6 = 37 - 31 = 43 - 37;
%C A270269 90 = 8101 - 8011 = 8191 - 8101.
%C A270269 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, ...}.
%C A270269 Examples:
%C A270269 a(3) + A087277(1) = 7 + 2 = 3^2;
%C A270269 a(6) + A087277(2) = 43 + 6 = 7^2;
%C A270269 a(9) + A087277(3) = 8191 + 90 = 91^2.
%H A270269 Michel Lagneau, <a href="/A270269/a270269.pdf">Illustration</a>
%p A270269 nn:=20000:T:=array(1..nn):a0:=1:kk:=0:
%p A270269 for p from 1 to nn do :
%p A270269    a1:=a0+floor(p/2):a0:=a1:
%p A270269     if  isprime(a1)
%p A270269      then
%p A270269      kk:=kk+1:T[kk]:=a1:
%p A270269      else
%p A270269     fi:
%p A270269   od:
%p A270269    for n from 1 to kk-2 do:
%p A270269     d1:=T[n+2]-T[n+1]:d2:=T[n+1]-T[n]:
%p A270269      if d1=d2
%p A270269      then
%p A270269       printf("%d %d %d \n", T[n], T[n+1], T[n+2]):
%p A270269       else
%p A270269      fi:
%p A270269    od:
%Y A270269 Cf. A087277, A172979.
%K A270269 nonn
%O A270269 1,1
%A A270269 _Michel Lagneau_, Mar 14 2016