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A192297
Lesser of pseudo twin primes to base 2.
2
561, 643, 645, 1103, 1905, 2465, 2699, 2819, 4369, 4371, 4679, 6599, 10259, 12799, 14489, 16703, 18719, 19949, 23001, 25759, 25761, 29339, 30119, 31607, 33151, 39863, 41039, 42797, 49139, 52631, 55243, 60701, 62743, 68099, 72883, 83663, 85487, 87249, 90749
OFFSET
1,1
COMMENTS
We call numbers {k,k+2} pseudo twin primes to base 2 if at least one of them is composite, while 2^(k-1) == 1 (mod k) and 2^(k+1) == 1 mod (k+2).
4369 is the only known term such that both k and k+2 are composite (cf. A173619). - Jianing Song, Nov 20 2021
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..15587 (terms below 10^12; terms 1..1000 from Alois P. Heinz)
FORMULA
2^(a(n) + 2) == 3*a(n) + 8 (mod a(n)*(a(n)+2)).
4*(2^(a(n)-1)-1) == -a(n)*((a(n)-1)/2) (mod a(n)*(a(n)+2)). - Davide Rotondo, Nov 07 2021
MAPLE
a:= proc(n) option remember; local k;
for k from 2+`if` (n=1, 1, a(n-1)) by 2 while
isprime(k) and isprime(k+2) or
(2&^(k-1) mod k)<>1 or (2&^(k+1) mod (k+2))<>1
do od; k
end:
seq (a(n), n=1..40); # Alois P. Heinz, Oct 13 2011
MATHEMATICA
fQ[n_] := (! PrimeQ[n] || ! PrimeQ[n + 2]) && PowerMod[2, n - 1, n] == 1 && PowerMod[2, n + 1, n + 2] == 1; Select[2 Range@ 32000 + 1, fQ] (* Robert G. Wilson v, Oct 11 2011 *)
PROG
(PARI) is(n)=Mod(2, n^2+2*n)^(n+2)==3*n+8 && (!isprime(n) || !isprime(n+2)) && n>1 \\ Charles R Greathouse IV, Dec 02 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 11 2011
STATUS
approved