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A160440 Smaller member of a pair (p,q) of cousin primes such that p and q are in different centuries.

%I A160440 #13 Mar 29 2023 09:01:49
%S A160440 97,397,499,1297,1597,1999,2797,3697,4999,6199,6997,7699,9199,10099,
%T A160440 10597,12097,13099,16699,18397,20899,21397,21499,21799,23197,23599,
%U A160440 25999,26497,27697,27799,27997,32299,32797,33199,34297,35797,38197,38299,39499,42697
%N A160440 Smaller member of a pair (p,q) of cousin primes such that p and q are in different centuries.
%C A160440 Sequence is probably infinite.
%C A160440 Dickson's conjecture implies there are infinitely many pairs of primes (100*k-3, 100*k+1) and infinitely many pairs of primes (100*k-1, 100*k+3). - _Robert Israel_, Mar 28 2023
%C A160440 It appears that every integer occurs as the difference round((a(n+1)-a(n))/100); all numbers 1..298 occur as these differences for a(n) < 1000000000. - _Hartmut F. W. Hoft_, May 18 2017
%H A160440 Robert Israel, <a href="/A160440/b160440.txt">Table of n, a(n) for n = 1..10000</a>
%F A160440 {A023200(n): [A023200(n)/100] <> [A046132(n)/100]}, where [..]=floor(..).
%e A160440 Cousin primes 1597 and 1601 are in successive (that is 16th and 17th) centuries.
%p A160440 R:= NULL: count:= 0:
%p A160440 for i from 1 while count < 100 do
%p A160440   if ((i mod 3 = 1) and isprime(100*i-3) and isprime(100*i+1)) then
%p A160440      R:= R, 100*i-3; count:= count+1
%p A160440   elif ((i mod 3 = 2) and isprime(100*i-1) and isprime(100*i+3)) then
%p A160440      R:= R, 100*i-1; count:= count+1
%p A160440 fi od:
%p A160440 R; # _Robert Israel_, Mar 28 2023
%t A160440 a160440[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[100, n, 100]], First[#]-Last[#]==4&]]
%t A160440 a160440[43000] (* data *) (* _Hartmut F. W. Hoft_, May 18 2017 *)
%Y A160440 Cf. A046132, A160370
%K A160440 nonn
%O A160440 1,1
%A A160440 _Ki Punches_, May 13 2009
%E A160440 Edited by _R. J. Mathar_, May 14 2009