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Emirps p such that both p and its digit reversal can be written as q*r+q+r where q and r are emirps.
1

%I #18 Sep 17 2022 14:11:25

%S 134999,999431,1383947,1903103,3013091,3626339,7282487,7493831,

%T 7842827,9336263,9366839,9386639,9562499,9942659,11230199,11370743,

%U 11394431,11650571,11769839,11884079,13182623,13413599,13449311,13611023,13683179,13881323,15123527,15788771,15925391,15934463,17505611

%N Emirps p such that both p and its digit reversal can be written as q*r+q+r where q and r are emirps.

%C Members of A352249 whose digit reversals are also in A352249.

%H Robert Israel, <a href="/A353031/b353031.txt">Table of n, a(n) for n = 1..1000</a>

%e a(6) = 3626339 is a term because 3626339 = 37*95429 + 37 + 95429, its digit reversal 9336263 = 97*95267 + 97 + 95267, and 3626339, 37, 95429, 97 and 95267 are all emirps.

%p revdigs:= proc(n) local L, i; L:= convert(n, base, 10); add(L[-i]*10^(i-1), i=1..nops(L)) end proc:

%p isemirp:= proc(p) local r;

%p if not isprime(p) then return false fi;

%p r:= revdigs(p);

%p r <> p and isprime(r)

%p end proc:

%p filter:= proc(p) local q,t,flag;

%p if not isprime(p) then return false fi;

%p q:= revdigs(p);

%p if q=p or not isprime(q) then return false fi;

%p flag:= false;

%p for t in select(`<`, numtheory:-divisors(p+1),floor(sqrt(p+1))) do

%p if isemirp(t-1) and isemirp((p+1)/t-1) then flag:= true; break fi

%p od;

%p if not flag then return false fi;

%p for t in select(`<`, numtheory:-divisors(q+1),floor(sqrt(q+1))) do

%p if isemirp(t-1) and isemirp((q+1)/t-1) then return true fi

%p od;

%p false

%p end proc:

%p p:= 2: R:= NULL: count:= 0:

%p while count < 40 do

%p p:= nextprime(p);

%p d:= ilog10(p);

%p p1:= floor(p/10^d);

%p if p1=2 then p:= nextprime(3*10^d)

%p elif member(p1,{4,5,6}) then p:= nextprime(7*10^d)

%p elif p1=8 then p:= nextprime(9*10^d)

%p fi;

%p if filter(p) then R:= R, p; count:= count+1 fi;

%p od:

%p R;

%Y Cf. A004086, A006567, A352249.

%K nonn,base

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Apr 18 2022