OFFSET
1,1
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
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:
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 14 2016
STATUS
approved