A353030 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.

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

Offset: 0

Author

J. M. Bergot and Robert Israel, Apr 18 2022

Keywords

• nonn
• base
• more

Comments

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.

From David A. Corneth, Jan 14 2023: (Start)

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

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

Examples

			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.
		

Links

• David A. Corneth, Upper bounds on a(0)..a(30).

Crossrefs

Cf. A006567, A352249.

Programs

Maple

revdigs:= proc(n) local L, i; L:= convert(n, base, 10); add(L[-i]*10^(i-1), i=1..nops(L)) end proc:
isemirp:= proc(p) local r;
   if not isprime(p) then return false fi;
   r:= revdigs(p);
   r <> p and isprime(r)
end proc:
g:= proc(n) local p,q, t,count;
  count:= 0;
  for t in select(`<`,numtheory:-divisors(n+1),floor(sqrt(n+1))) do
    if isemirp(t-1) and isemirp((n+1)/t-1) then
       count:= count+1;
    fi
  od;
  count
end proc:
V:= Array(0..6): vcount:= 0:
p:= 2:
while vcount < 7 do
  p:= nextprime(p);
  d:= ilog10(p);
  p1:= floor(p/10^d);
  if p1=2 then p:= nextprime(3*10^d)
  elif member(p1,{4,5,6}) then p:= nextprime(7*10^d)
  elif p1=8 then p:= nextprime(9*10^d)
  fi;
  if isemirp(p) then
    v:= g(p);
    if V[v] = 0 then vcount:= vcount+1; V[v]:= p; fi;
  fi
od:
convert(V,list);

Extensions

a(7) from David A. Corneth, Jan 14 2023