A216166 Composite numbers and 1 which yield a prime whenever a 3 is inserted anywhere in them (including at the beginning or end).

1, 121, 343, 361, 533, 637, 793, 889, 943, 1183, 3013, 3223, 3353, 3403, 3757, 3827, 3893, 4313, 4543, 4963, 8653, 10423, 14257, 20339, 23083, 23419, 30917, 33031, 33101, 33323, 33433, 33701, 33821, 34333, 34393, 35453, 36437, 36533, 39137, 39247, 42869, 43337

Offset: 1

Author

Paolo P. Lava, Sep 03 2012

Keywords

• nonn
• base

Examples

			3827 is not prime but 38273, 38237, 38327 and 33827 are all primes.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A068673, A068674, A068677, A068679, A069246, A215417, A215419-A215421, A216165, A216167-A216169.

Programs

Magma

[n: n in [1..50000] | not IsPrime(n) and forall{m: t in [0..#Intseq(n)] | IsPrime(m) where m is (Floor(n/10^t)*10+3)*10^t+n mod 10^t}]; // Bruno Berselli, Sep 03 2012

Maple

with(numtheory);
A216166:=proc(q,x)
local a,b,c,i,n,ok;
for n from 1 to q do
if not isprime(n) then
  a:=n; b:=0; while a>0 do b:=b+1; a:=trunc(a/10); od; a:=n; ok:=1;
  for i from 0 to b do c:=a+9*10^i*trunc(a/10^i)+10^i*x;
    if not isprime(c) then ok:=0; break; fi;
  od;
  if ok=1 then print(n); fi;
fi;
od; end:
A216166(1000,3);

Mathematica

ap3Q[n_]:=CompositeQ[n]&&AllTrue[FromDigits/@Table[Insert[ IntegerDigits[ n],3,k],{k,IntegerLength[n]+1}],PrimeQ]; Join[{1},Select[Range[ 44000], ap3Q]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 25 2020 *)