A136720 -id:A136720 - cp's OEIS frontend

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

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

nonn

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

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

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

A136720 is a subsequence.

Cf. A255581, A358742, A358744.

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;

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

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

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

A136721 Prime quadruples: 3rd term.

Original entry on oeis.org

11, 17, 107, 197, 827, 1487, 1877, 2087, 3257, 3467, 5657, 9437, 13007, 15647, 15737, 16067, 18047, 18917, 19427, 21017, 22277, 25307, 31727, 34847, 43787, 51347, 55337, 62987, 67217, 69497, 72227, 77267, 79697, 81047, 82727, 88817, 97847

Offset: 1

Enoch Haga, Jan 18 2008

Primes p such that p-6, p-4, and p+2 are prime. Apart from the first term, a(n) = 17 (mod 30).

The members of each quadruple are twin primes when they are 1st and 2nd terms and when 3rd and 4th terms. When they are 2nd and 3rd terms they differ by 4.

			The four terms in the first quadruple are 5,7,11,13 and in the 2nd 11,13,17,19. The four terms or members of each set must be simultaneously prime.

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

Cf. A007530, A090258, A136720.

p2:= 0: p3:= 0: p4:= 0:
Res:= NULL: count:= 0:
while count < 100 do
  p1:= p2; p2:= p3; p3:= p4;
  p4:= nextprime(p4);
  if [p2-p1, p3-p2, p4-p3] = [2, 4, 2] then
     count:= count+1; Res:= Res, p3
  fi
od:
Res; # Robert Israel, Oct 11 2019

lst={};Do[p0=Prime[n];If[PrimeQ[p2=p0+2], If[PrimeQ[p6=p0+6], If[PrimeQ[p8=p0+8], AppendTo[lst, p6]]]], {n, 12^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)

a(n) = A007530(n)+6 = A136720(n)+4 = A090258(n)-2. - Robert Israel, Oct 11 2019

Edited by Charles R Greathouse IV, Oct 11 2009

cp's OEIS Frontend

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

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