A216167 Composite numbers which yield a prime whenever a 5 is inserted anywhere in them, excluding at the end.

9, 21, 57, 63, 69, 77, 87, 93, 153, 231, 381, 407, 413, 417, 501, 531, 581, 651, 669, 741, 749, 783, 791, 987, 1241, 1551, 1797, 1971, 2189, 2981, 3381, 3419, 3591, 3951, 4083, 4503, 4833, 4949, 4959, 5049, 5117, 5201, 5229, 5243, 5529, 5547, 5603, 5691, 5697

Offset

1,1

Links

Michael S. Branicky, Table of n, a(n) for n = 1..1923 (terms 1..300 from Paolo P. Lava)

Example

4083 is not prime but 40853, 40583, 45083 and 54083 are all primes.

Maple

with(numtheory);
A216167:=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 1 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:
A216167(1000, 5);

Mathematica

Select[Range[6000], CompositeQ[#]&&AllTrue[FromDigits/@Table[Insert[IntegerDigits[#], 5, p], {p, IntegerLength[#]}], PrimeQ]&] (* Harvey P. Dale, Oct 02 2022 *)

Prog

(Magma) [n: n in [1..6000] | not IsPrime(n) and forall{m: t in [1..#Intseq(n)] | IsPrime(m) where m is (Floor(n/10^t)*10+5)*10^t+n mod 10^t}]; // Bruno Berselli, Sep 03 2012
(Python)
from sympy import isprime
def ok(n):
    if n < 2 or n%10 not in {1, 3, 7, 9} or isprime(n): return False
    s = str(n)
    return all(isprime(int(s[:i] + '5' + s[i:])) for i in range(len(s)))
print(list(filter(ok, range(5698)))) # Michael S. Branicky, Sep 21 2021

Crossrefs

Cf. A068673, A068674, A068677, A068679, A069246, A215417, A215419-A215421, A216165, A216166, A216168, A216169.

Subsequence of A053795.

Sequence in context: A134717 A216980 A147169 * A246327 A320896 A127989

Adjacent sequences: A216164 A216165 A216166 * A216168 A216169 A216170

Keyword

nonn,look,base

Author

Paolo P. Lava, Sep 03 2012

Status

approved