A358743 First of three consecutive primes p,q,r such that p+q-r is prime.

7, 11, 13, 17, 19, 29, 41, 43, 47, 59, 79, 101, 103, 107, 113, 137, 139, 163, 181, 193, 227, 229, 239, 257, 269, 281, 283, 311, 317, 359, 379, 397, 419, 421, 439, 461, 487, 491, 503, 521, 547, 569, 577, 599, 647, 659, 683, 691, 701, 709, 761, 811, 823, 857, 863, 881, 883, 887, 919, 983, 1019

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 29 2022

Keywords

• nonn

Comments

p+q-r is near (and less than) p and odd (for p > 2), so heuristically we would expect it to be prime about 2/log p of the time, yielding around 2x/log^2 x terms up to x. (A more careful analysis of small primes could yield a slightly different leading constant.) - Charles R Greathouse IV, Nov 29 2022

Examples

			a(3) = 13 is a prime because 13, 17, 19 are three consecutive primes with 13 + 17 - 19 = 11 prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

A136720 is a subsequence.

Cf. A255581, A358742, A358744.

Programs

Maple

R:= NULL: count:= 0: q:= 2: r:= 3:
while count < 100 do
  p:= q; q:= r; r:=nextprime(r);
  if isprime(p+q-r) then count:= count+1; R1:= R1,p fi;
od:
R;

Mathematica

Select[Partition[Prime[Range[180]], 3, 1], PrimeQ[#[[1]] + #[[2]] - #[[3]]] &][[;; , 1]] (* Amiram Eldar, Nov 29 2022 *)

PARI

list(lim)=my(v=List(),p=7,q=11); forprime(r=13,nextprime(nextprime(lim\1+1)+1), if(isprime(p+q-r), listput(v,p)); p=q; q=r); Vec(v) \\ Charles R Greathouse IV, Nov 29 2022

Python

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        if isprime(p+q-r): yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 61))) # Michael S. Branicky, Nov 29 2022