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A100873 Pseudotwinprimes p+2 for primes p such that p+2 divides p^(p+2)+2 and p+2 is composite. 2
645, 1105, 2701, 2821, 4681, 6601, 10261, 12801, 14491, 16705, 18721, 19951, 25761, 29341, 30121, 31609, 33153, 39865, 41041, 42799, 49141, 52633, 55245, 62745, 68101, 72885, 83665, 85489, 90751, 104653, 107185, 129889, 129921 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture 1: If p and p+2 are prime (twin primes), then p+2 divides p^(p+2)+2. Compared to the 1517 twin primes less than 130000, there were 33 pseudoprime occurrences. Conjecture 2: If for a randomly chosen prime p, p+2 divides p^(p+2)+2, then there is a greater than 98% chance that p and p+2 are twin primes. The sequence also contains several Carmichael numbers. In addition, If we relax the condition that p is prime or just odd, we get A001567 341, 561, 645, 1105, 1387, 1729, 1905, 2047, ... (Sarrus numbers).

Proof 1: p^(p+2) == p == -2 (mod p+2). - Thomas Ordowski, Dec 19 2013

Base-2 pseudoprimes k such that k-2 is prime. - Charlie Neder, Mar 04 2019

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

For primes p if p+2 divides p^(p+2) + 2 then p+2 is likely to be prime. If p+2 is composite, then p+2 is a pseudotwinprime.

EXAMPLE

For prime p = 643, 645 divides 643^(645) + 2 and 645 is composite.

MATHEMATICA

Select[Range[10^5], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 && PrimeQ[# - 2] &] (* Amiram Eldar, Jun 28 2019 *)

PROG

(PARI) pitwins(n) = { local(c, x); c=0; forprime(x=3, n, if(isprime(x+2), c++) ); return(c) } \\ The number of twin prime pairs <= n.

twtotwp2(n1, n2, k=2) = { local(x, y, x2, c); c=0; forprime(x=n1, n2, x2=x+2; y=x^x2+k; if(y%x2==0 && !isprime(x2), c++; print1(x+2", "); ); ); /* print(); print(c", "pitwin(n2)); */}

CROSSREFS

Sequence in context: A304607 A168626 A216023 * A227136 A216364 A063844

Adjacent sequences:  A100870 A100871 A100872 * A100874 A100875 A100876

KEYWORD

hard,nonn

AUTHOR

Cino Hilliard, Jan 09 2005

STATUS

approved

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Last modified January 23 17:09 EST 2022. Contains 350514 sequences. (Running on oeis4.)