A216168 -id:A216168 - cp's OEIS frontend

A216169 Composite numbers > 9 which yield a prime whenever a 0 is inserted between any two digits.

49, 119, 121, 133, 161, 169, 203, 253, 299, 301, 319, 323, 403, 407, 473, 493, 511, 539, 551, 581, 611, 667, 679, 713, 869, 901, 913, 943, 1007, 1067, 1079, 1099, 1211, 1273, 1691, 1729, 1799, 1909, 2021, 2047, 2101, 2117, 2359, 2407, 2533, 2717, 2759, 2899

Offset: 1

Paolo P. Lava, Sep 03 2012

			2359 is not prime but 23509, 23059 and 20359 are all primes.

Paolo P. Lava and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1500 terms from Paolo P. Lava)

Subset of composite numbers in A164329. - M. F. Hasler, May 10 2018

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

A216169:=proc(q,x)
local a,b,c,i,n,ok;
for n from 10 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-1 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: A216169(1000,0);

Select[Range[10,3000],CompositeQ[#]&&AllTrue[Table[FromDigits[ Insert[ IntegerDigits[ #],0,n]],{n,2,IntegerLength[#]}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2018 *)

is(n, L=logint(n+!n, 10)+1, P)={!isprime(n) && !for(k=1, L-1, isprime([10*P=10^(L-k),1]*divrem(n, P))||return) && n>9} \\ M. F. Hasler, May 10 2018

Name edited by M. F. Hasler, May 10 2018

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

Original entry on oeis.org

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

Paolo P. Lava, Sep 03 2012

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

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

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

Subsequence of A053795.

[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

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);

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

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

cp's OEIS Frontend

A216169 Composite numbers > 9 which yield a prime whenever a 0 is inserted between any two digits.

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