A341217 a(k) is the lesser of the first pair of twin primes that starts a chain of k pairs of twin primes (p(1),p(1)+2), ..., (p(k),p(k)+2) where p(j+1) = 5*p(j)+4.

3, 11, 5, 1720949, 22362444257, 57703877539769

Offset: 1

Author

J. M. Bergot and Robert Israel, Feb 06 2021

Keywords

• nonn
• more

Comments

a(5) > 10^8 if it exists.

Examples

			3 and 5 are twin primes, but 5*3+4 = 19 and 21 are not twin primes.
11 and 13 are twin primes, 5*11+4 = 59 and 61 are twin primes, but 5*59+4 = 299 and 301 are not twin primes.
5 and 7 are twin primes, 5*5+4 = 29 and 31 are twin primes, 5*29+4 = 149 and 151 are twin primes, but 5*149+4 = 749 and 751 are not twin primes.
1720949 and 1720951 are twin primes, 5*1720949+4 = 8604749 and 8604751 are twin primes, 5*8604749+4 = 43023749 and 43023751 are twin primes,
5*8604749+4 = 215118749 and 215118751 are twin primes, 5*1075593749+4 = 1075593749 and 1075593751 are not twin primes.
		

Crossrefs

Cf. A001359, A174922.

Programs

Maple

V:= [3,0,0,0]:
count:= 1:
for p from 5 by 6 while count < 4 do
  if isprime(p) and isprime(p+2) then
    ct:= 1:
    q:= p;
    do
      q:= 5*q+4;
      if not (isprime(q) and isprime(q+2)) then break fi;
      ct:= ct+1;
    od;
    if V[ct] = 0 then V[ct]:= p; count:= count+1; fi;
  fi
od:
V;

Extensions

a(5) from Martin Ehrenstein, Feb 07 2021

a(6) from Martin Ehrenstein, Feb 10 2021