A353030 - OEIS

%I #24 Jan 15 2023 09:52:06

%S 13,1439,100799,3548879,14061599,38342303,120355199,12555446399

%N a(n) is the first emirp p such that there are exactly n unordered pairs (q,r) of emirps with p = q*r + q + r.

%C a(n) is the first prime p such that the digit-reversal rev(p) of p is a prime and there are exactly n pairs (q,r) of primes such that q < r, rev(q) and rev(r) are primes, and p = q*r + q + r.

%C From _David A. Corneth_, Jan 14 2023: (Start)

%C a(8) <= 121347071999, a(9) <= 195271876799, a(10) <= 10175362797599, a(11) <= 17482966300799.

%C For n >= 2, n == 3 (mod 4) and (n + 1)/4 has at least 2*n divisors. (End)

%H David A. Corneth, <a href="/A353030/a353030.gp.txt">Upper bounds on a(0)..a(30).</a>

%e a(3) = 3548879 because 3548879 = 17*197159 + 17 + 197159 = 359*9857 + 359 + 9857 = 953*3719 + 953 + 3719 and 3548879, 17, 197159, 359, 9857, 953, 3719 are 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 g:= proc(n) local p,q, t,count;

%p   count:= 0;

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

%p     if isemirp(t-1) and isemirp((n+1)/t-1) then

%p        count:= count+1;

%p     fi

%p   od;

%p   count

%p end proc:

%p V:= Array(0..6): vcount:= 0:

%p p:= 2:

%p while vcount < 7 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 isemirp(p) then

%p     v:= g(p);

%p     if V[v] = 0 then vcount:= vcount+1; V[v]:= p; fi;

%p   fi

%p od:

%p convert(V,list);

%Y Cf. A006567, A352249.

%K nonn,base,more

%O 0,1

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

%E a(7) from _David A. Corneth_, Jan 14 2023