A350884 Emirps p such that, if q is the next emirp after p, p*q mod (p+q) and floor(p*q/(p+q)) are both emirps.

15733, 15803, 18413, 19037, 37243, 75913, 157363, 371057, 393919, 396509, 705169, 722983, 740477, 794141, 1857599, 1858093, 1858643, 1865491, 1918529, 1922351, 1950989, 3002977, 3006551, 3007723, 3127139, 3234857, 3266369, 3444017, 3548891, 3614339, 3658981, 3687127, 3734657, 3763567, 3807173

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..2000

Example

a(3) = 18413 = p is a term because it is an emirp (18413 and 31481 being distinct primes), the next emirp is q = 18427, and (p*q) mod (p+q) = 36791 and floor((p*q)/(p+q)) = 9209 are emirps.

Maple

rev:= proc(n) local L, i;
  L:= convert(n, base, 10);
  add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
isemirp:= proc(n) local r;
   if not isprime(n) then return false fi;
   r:= rev(n);
   r <> n and isprime(r)
end proc:
R:= NULL: count:= 0:
p:= 0:
for d from 1 while count < 40 do
for i in [1, 3, 7, 9] do
   for j from 1 to 10^d-1 by 2 while count < 40 do
     q:= i*10^d+j;
     if isemirp(q) then
        s:= p+q;
        t:= p*q;
        if isemirp(t mod s) and isemirp(floor(t/s)) then
           count:= count+1; R:= R, p;
        fi;
        p:= q;
     fi;
od od od;
R;

Mathematica

emirpQ[p_] := (q = IntegerReverse[p]) != p && And @@ PrimeQ[{p, q}]; nextEmirp[p_] := Module[{k = NextPrime[p]}, While[(q = IntegerReverse[k]) == k || ! PrimeQ[q], k = NextPrime[k]]; k]; seqQ[p_] := emirpQ[p] && Module[{q = nextEmirp[p]}, And @@ emirpQ /@ {Mod[p*q, p + q], Floor[p*q/(p + q)]}]; Select[Range[2*10^6], seqQ] (* Amiram Eldar, Jan 21 2022 *)

Crossrefs

Cf. A006567, A346147.

Sequence in context: A298758 A205289 A205897 * A190834 A183966 A187031

Adjacent sequences: A350881 A350882 A350883 * A350885 A350886 A350887

Keyword

nonn,base

Author

J. M. Bergot and Robert Israel, Jan 20 2022

Status

approved