A217046 -id:A217046 - cp's OEIS frontend

A217047 Primes that remain prime when a single "8" digit is inserted between any two adjacent digits.

11, 23, 47, 83, 131, 173, 179, 233, 353, 389, 521, 569, 641, 683, 839, 887, 911, 971, 983, 1229, 1289, 1913, 2087, 2663, 2837, 2879, 3329, 3671, 3677, 3803, 3821, 4259, 4409, 4817, 4871, 4889, 5237, 5477, 5693, 6449, 6581, 6863, 7283, 7487, 7583, 7823, 7853

Offset: 1

Paolo P. Lava, Sep 25 2012

These numbers are either isolated primes or the smaller of a pair of twin primes. - Davide Rotondo, Mar 11 2025

			325421 is prime and also 3254281, 3254821, 3258421, 3285421 and 3825421.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044-A217046, A217062-A217065.

[p: p in PrimesInInterval(11,8000) | forall{m: t in [1..#Intseq(p)-1] | IsPrime(m) where m is (Floor(p/10^t)*10+8)*10^t+p mod 10^t}]; // Bruno Berselli, Sep 26 2012

A217044:=proc(q,x) local a,b,c,d,i,k,n,ok,v; v:=[]; a:=10;
for n from 1 to q do a:=nextprime(a); d:=length(a); ok:=1;
for k from 1 to d-1 do b:=a mod 10^k; c:=trunc(a/10^k); i:=x*10^k+b; i:=c*10^length(i)+i;
if not isprime(i) then ok:=0; break; fi; od; if ok=1 then v:=[op(v),a]; fi; od; op(v); end:
A217044(10^3,8);

is(n)=my(v=concat([""],digits(n)));for(i=2,#v-1,v[1]=Str(v[1], v[i]); v[i]=8;if(i>2,v[i-1]="");if(!isprime(eval(concat(v))), return(0)));isprime(n) \\ Charles R Greathouse IV, Sep 25 2012

from sympy import isprime, primerange
def ok(p):
    if p < 10: return False
    s = str(p)
    return all(isprime(int(s[:i] + "8" + s[i:])) for i in range(1, len(s)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(7854)) # Michael S. Branicky, Nov 23 2021

A217045 Primes that remain prime when a single "4" digit is inserted between any two adjacent decimal digits.

Original entry on oeis.org

19, 37, 43, 61, 67, 73, 97, 109, 199, 211, 223, 241, 349, 409, 421, 457, 463, 541, 571, 751, 757, 823, 991, 1033, 1087, 1321, 1423, 1447, 1543, 2749, 3361, 3469, 3499, 3847, 4111, 4273, 4483, 5059, 5437, 5443, 5449, 6373, 6709, 6793, 7687, 8089, 8221, 8443

Offset: 1

Paolo P. Lava, Sep 25 2012

			87697 is prime and also 876947, 876497, 874697 and 847697.

Paolo P. Lava, Table of n, a(n) for n = 1..141

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044, A217046, A217047, A217062-A217065

with(numtheory);
A217045:=proc(q,x)
local a,b,c,i,n,ok;
for n from 5 to q do
a:=ithprime(n); b:=0;
while a>0 do b:=b+1; a:=trunc(a/10); od; a:=ithprime(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(ithprime(n)); fi;
od; end:
A217045(100000,4)

Select[Prime[Range[5,1500]],AllTrue[Table[FromDigits[Insert[ IntegerDigits[ #],4,n]],{n,2,IntegerLength[#]}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 04 2017 *)

is(n)=my(v=concat([""], digits(n))); for(i=2, #v-1, v[1]=Str(v[1], v[i]); v[i]=4; if(i>2, v[i-1]=""); if(!isprime(eval(concat(v))), return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 26 2012

A349636 Primes that remain prime when a single "1" digit is inserted between any two adjacent digits.

Original entry on oeis.org

13, 31, 37, 67, 79, 103, 109, 151, 163, 181, 193, 211, 241, 367, 457, 547, 571, 601, 613, 631, 709, 787, 811, 1117, 1213, 1831, 2017, 2683, 3019, 3319, 3391, 3511, 3517, 3607, 4519, 4999, 6007, 6121, 6151, 6379, 6673, 6871, 6991, 8293, 11119, 11317, 11467

Offset: 1

Michael S. Branicky, Nov 23 2021

			37 and 317 are prime; 2683 is prime, as are 21683, 26183, and 26813.

Martin Ehrenstein, Table of n, a(n) for n = 1..2004

The terms of A069246 > 10 are a subsequence.
Cf. A215417 (same with 0), A217044 (2), A217045 (4), A217046 (6), A217047 (8), A217062 (9), A217063 (3), A217064 (5), A217065 (7).
Subsequence of A002476.

Select[Prime@Range[5,1500],(p=#;And@@PrimeQ[FromDigits/@(Insert[IntegerDigits@p,1,#]&/@Range[2,IntegerLength@p])])&] (* Giorgos Kalogeropoulos, Nov 23 2021 *)

from sympy import isprime, primerange
def ok(p):
    if p < 10: return False
    s = str(p)
    return all(isprime(int(s[:i] + "1" + s[i:])) for i in range(1, len(s)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(12000)) # Michael S. Branicky, Nov 23 2021

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A217047 Primes that remain prime when a single "8" digit is inserted between any two adjacent digits.

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A217045 Primes that remain prime when a single "4" digit is inserted between any two adjacent decimal digits.

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Programs

Maple

Mathematica

PARI

A349636 Primes that remain prime when a single "1" digit is inserted between any two adjacent digits.

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Author

Keywords

Examples

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Programs

Mathematica

Python