A157750 Lesser of two consecutive primes p,q such that q^2 - p^2 + 1 = the square of a prime.

5, 11, 13, 19, 29, 41, 43, 71, 103, 151, 181, 229, 239, 349, 419, 461, 463, 491, 571, 859, 1069, 1429, 1483, 1583, 1721, 2549, 2969, 3079, 3191, 3319, 3331, 4003, 7177, 7193, 7309, 7873, 8009, 8161, 8849, 9127, 9283, 10729, 11779, 13567, 13693, 15809, 15959

Offset

1,1

Comments

One could generate a larger sequence using any three primes p,q,r such that p^2 + 1 = q^2 + r^2. One could consider these "almost prime Pythagorean triangles."

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

For the consecutive pair (19,23), 23^2 - 19^2 + 1 = 169 = 13^2; thus 19 is in the sequence.

Maple

a := proc (n) if isprime(sqrt(nextprime(ithprime(n))^2-ithprime(n)^2+1)) = true then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 2000); # Emeric Deutsch, Mar 07 2009

Mathematica

ltcpQ[{a_, b_}]:=PrimeQ[Sqrt[b^2-a^2+1]]; Select[Partition[ Prime[ Range[ 2000]], 2, 1], ltcpQ][[All, 1]] (* Harvey P. Dale, Jul 23 2021 *)

Prog

(PARI) list(lim)=my(v=List(), p=2, t); forprime(q=3, nextprime(lim\1+1), if(issquare(q^2-p^2+1, &t)&&isprime(t), listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jan 31 2017

Crossrefs

Cf. A001481.

Sequence in context: A040144 A019395 A045448 * A359501 A045449 A296859

Adjacent sequences: A157747 A157748 A157749 * A157751 A157752 A157753

Keyword

nonn

Author

J. M. Bergot, Mar 05 2009

Extensions

More terms from Klaus Brockhaus, Mar 05 2009

More terms from Emeric Deutsch, Mar 07 2009

Status

approved