A350884 - OEIS

%I #10 Jan 25 2022 04:21:36

%S 15733,15803,18413,19037,37243,75913,157363,371057,393919,396509,

%T 705169,722983,740477,794141,1857599,1858093,1858643,1865491,1918529,

%U 1922351,1950989,3002977,3006551,3007723,3127139,3234857,3266369,3444017,3548891,3614339,3658981,3687127,3734657,3763567,3807173

%N 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.

%H Robert Israel, <a href="/A350884/b350884.txt">Table of n, a(n) for n = 1..2000</a>

%e 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.

%p rev:= proc(n) local L,i;

%p   L:= convert(n,base,10);

%p   add(L[-i]*10^(i-1),i=1..nops(L))

%p end proc:

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

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

%p    r:= rev(n);

%p    r <> n and isprime(r)

%p end proc:

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

%p p:= 0:

%p for d from 1 while count < 40 do

%p for i in [1,3,7,9] do

%p    for j from 1 to 10^d-1 by 2 while count < 40 do

%p      q:= i*10^d+j;

%p      if isemirp(q) then

%p         s:= p+q;

%p         t:= p*q;

%p         if isemirp(t mod s) and isemirp(floor(t/s)) then

%p            count:= count+1; R:= R, p;

%p         fi;

%p         p:= q;

%p      fi;

%p od od od;

%p R;

%t 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 *)

%Y Cf. A006567, A346147.

%K nonn,base

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Jan 20 2022