A087525 Primes p with the property that p-q does not divide p+q for all primes q < p.

5, 7, 7, 11, 11, 11, 11, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41

Offset

1,1

Comments

Theorem: If z>y are primes then z-y divides z+y iff z=y+2. Proof: Let (1) z+y = (z-y)k for some integer k (2) z=y+2m. We must add multiples of 2 to y in order to avoid z=even. Substituting (2) into (1) we get y+2m + y = (y+2m-y)k 2y+2m = 2mk y+m = mk y/m + 1 = k for k to be an integer m must be 1 or y. If m = y then k=2 and z+y = 2*(z-y) z+y = 2z-2y 3y = z contradicting z > y. Therefore m=1 and z = y+2m = y+2 as desired. If z-y does divide z+y we have z the sequence of upper bound twin primes.

Links

Table of n, a(n) for n=1..70.

Prog

(PARI) zmy(n) = { forprime(z=1, n, forprime(y=1, z-1, v1=z-y; v2=z+y; if(v2%v1<>0, print1(z", ")) \ if(v2%v1==0, print1(z", ")) yields sequence of twin primes ) ) }

Crossrefs

Sequence in context: A244621 A201459 A053672 * A078496 A114521 A258653

Adjacent sequences: A087522 A087523 A087524 * A087526 A087527 A087528

Keyword

easy,nonn

Author

Cino Hilliard, Oct 23 2003

Extensions

Corrected by Franklin T. Adams-Watters and T. D. Noe, Oct 25 2006

Status

approved