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
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 1..2000
Programs
-
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 *)