A069246 -id:A069246 - cp's OEIS frontend

A247069 a(n) = number of distinct primes obtained when inserting 1 anywhere in A069246(n).

2, 2, 2, 2, 3, 3, 2, 2, 3, 4, 3, 3, 3, 2, 2, 3, 4, 4, 2, 3, 4, 3, 3, 4, 3, 5, 5, 5, 5, 4, 4, 4, 6, 4, 5, 5, 5, 6, 5, 4, 6, 6, 5, 4, 5, 6, 7, 6, 5, 5, 4, 5, 5, 6, 3, 5, 6, 5, 6, 5, 6, 3, 6, 7, 5, 5, 4, 5, 5, 5, 6, 4, 7, 4, 4, 6, 6, 4, 5, 7, 3, 3, 3, 4, 3, 5, 2, 3, 5, 5, 6, 7, 6, 4, 7, 5, 6, 5, 7, 3, 8

Offset: 1

Zak Seidov, Nov 17 2014

Among the first 145 terms the largest is a(140)=9 because inserting 1 in 12 places into A069246(140)=18064911343, we get only 9 distinct primes (in the order of their appearances): 118064911343, 181064911343, 180164911343, 180614911343, 180641911343, 180649111343, 180649113143, 180649113413, 180649113431.

			a(7)=2 because inserting 1 in 4 possible places into A069246(7)=151 we get only 2 distinct primes, 1151 and 1511.

Zak Seidov, Table of n, a(n) for n = 1..145

Cf. A069246.

A215417 Primes that remain prime when a single zero digit is inserted between any two adjacent digits.

Original entry on oeis.org

11, 13, 17, 19, 37, 41, 53, 59, 61, 67, 71, 79, 89, 97, 109, 113, 131, 149, 191, 197, 227, 239, 269, 281, 283, 337, 367, 379, 383, 401, 421, 449, 457, 499, 503, 509, 587, 607, 673, 701, 719, 727, 739, 757, 809, 811, 887, 907, 929, 991, 1009, 1061, 1093, 1103

Offset: 1

Paolo P. Lava, Aug 10 2012

			399617 is prime and also 3996107, 3996017, 3990617, 3909617, 3099617.

Giovanni Resta, Table of n, a(n) for n = 1..4300 (terms a(1)-a(372) from Paolo P. Lava, terms a(373)-a(700) from Vincenzo Librandi)

Cf. A159236, A069246, A215419-A215421.

A215417:=proc(q)
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); if not isprime(c) then ok:=0; break; fi;
  od;
  if ok=1 then print(ithprime(n)); fi;
od; end:
A215417(1000);

Select[Prime[Range[5,200]],And@@PrimeQ[Table[FromDigits[Insert[ IntegerDigits[ #],0,n]],{n,2,IntegerLength[#]}]]&] (* Harvey P. Dale, Feb 23 2014 *)

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

A215419 Primes that remain prime when a single digit 3 is inserted between any two consecutive digits or as the leading or trailing digit.

Original entry on oeis.org

7, 11, 17, 31, 37, 73, 271, 331, 359, 373, 673, 733, 1033, 2297, 3119, 3461, 3923, 5323, 5381, 5419, 6073, 6353, 9103, 9887, 18289, 23549, 25349, 31333, 32933, 33349, 35747, 37339, 37361, 37489, 47533, 84299, 92333, 93241, 95093, 98491, 133733, 136333, 139333

Offset: 1

Paolo P. Lava, Aug 10 2012

			18289 is prime and also 182893, 182839, 182389, 183289, 138289, 318289.

Robert Israel, Table of n, a(n) for n = 1..150

Cf. A215417, A069246, A215420, A215421

filter:= proc(n) local L,d,k,M;
if not isprime(n) then return false fi;
L:= convert(n,base,10);
d:= nops(L);
for k from 0 to d do
   M:= [seq(L[i],i=1..k),3,seq(L[i],i=k+1..d)];
   if not isprime(add(M[i]*10^(i-1),i=1..d+1)) then return false fi;
od;
true
end proc;
select(filter, [seq(i,i=3..2*10^5,2)]); # Robert Israel, Oct 09 2017

ins@n_:=Insert[IntegerDigits@n,3,#]&/@Range@(IntegerLength@n+1);
Cases[{#,FromDigits@#&/@ins@#}&/@ Cases[Range[11,70000],?PrimeQ], {,{?PrimeQ..}}][[All,1]] (* _Hans Rudolf Widmer, Dec 21 2023 *)

A068679 Numbers which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).

Original entry on oeis.org

1, 3, 7, 13, 31, 49, 63, 81, 91, 99, 103, 109, 117, 123, 151, 181, 193, 213, 231, 279, 319, 367, 427, 459, 571, 601, 613, 621, 697, 721, 801, 811, 951, 987, 1113, 1117, 1131, 1261, 1821, 1831, 1939, 2101, 2149, 2211, 2517, 2611, 3151, 3219, 4011, 4411, 4519, 4887, 5031, 5361, 6231, 6487, 6871, 7011, 7209, 8671, 9141, 9801, 10051

Offset: 1

Amarnath Murthy, Mar 02 2002

If R(p) = (10^p-1)/9 is a prime then (10^(p-1)-1)/9 belongs to this sequence.

			123 belongs to this sequence as the numbers 1123, 1213, 1231 obtained by inserting a 1 in all possible ways are all primes.

Giovanni Resta, Table of n, a(n) for n = 1..3314 (terms < 2*10^13, first 1929 terms from Chai Wah Wu)
C. Caldwell, Prime Pages

Cf. A068673, A068674, A068677, A069246.

d[n_]:=IntegerDigits[n]; ins[n_]:=FromDigits/@Table[Insert[d[n],1,k],{k,Length[d[n]]+1}]; Select[Range[10060],And@@PrimeQ/@ins[#] &] (* Jayanta Basu, May 20 2013 *)
Select[Range[11000],AllTrue[FromDigits/@Table[Insert[ IntegerDigits[ #],1,n],{n,IntegerLength[#]+1}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 16 2020 *)

from sympy import isprime
A068679_list, n = [], 1
while len(A068679_list) < 1000:
    if isprime(10*n+1):
        s = str(n)
        for i in range(len(s)):
            if not isprime(int(s[:i]+'1'+s[i:])):
                break
        else:
            A068679_list.append(n)
    n += 1 # Chai Wah Wu, Oct 02 2019

More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Apr 11 2002

More terms from Vladeta Jovovic, Apr 16 2002

A164329 Numbers which yield a prime whenever a zero is inserted between any two digits.

Original entry on oeis.org

11, 13, 17, 19, 37, 41, 49, 53, 59, 61, 67, 71, 79, 89, 97, 109, 113, 119, 121, 131, 133, 149, 161, 169, 191, 197, 203, 227, 239, 253, 269, 281, 283, 299, 301, 319, 323, 337, 367, 379, 383, 401, 403, 407, 421, 449, 457, 473, 493, 499, 503, 509, 511, 539, 551

Offset: 1

Farideh Firoozbakht, Sep 22 2009

Single-digit numbers 0, ..., 9 seem to be excluded but would satisfy the condition voidly. - M. F. Hasler, May 10 2018

			998471 is in the sequence because all the five numbers 9098471, 9908471, 9980471, 9984071 and 9984701 are primes.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A216169 (subset of composite terms), A215417 (subset of primes), A159236 (0 is inserted between all digits).
Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).
Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).
Cf. A069833 (9 is prefixed, appended or inserted anywhere), A215421 (primes among these).
Cf. A158232 (13 is prefixed or appended).

f[n_]:=(r=IntegerDigits[n];l=Length[r];For[k=2,PrimeQ[FromDigits[Insert
[r,0,k]]],k++ ];If[k==l+1,n,0]);Select[Range[11,560],f[ # ]>0&]

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

Erroneous comment and cross-references deleted by M. F. Hasler, May 10 2018

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

Original entry on oeis.org

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

A069833 Prefixing, suffixing or inserting a 9 in the number anywhere gives a prime.

Original entry on oeis.org

7, 19, 37, 41, 91, 199, 209, 239, 311, 539, 587, 661, 749, 923, 931, 941, 967, 1009, 1079, 1139, 1997, 2717, 2959, 3971, 3979, 4559, 4993, 4999, 5393, 5629, 5651, 6401, 6739, 6911, 8213, 8491, 8939, 9109, 9397, 9607, 9679, 9829, 11089, 11227, 13943

Offset: 1

Amarnath Murthy, Apr 14 2002

Giovanni Resta, Table of n, a(n) for n = 1..2128 (terms < 10^13, first 1175 terms from Chai Wah Wu)

Cf. A215421 (subsequence of primes).
Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).
Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).
Cf. A158232 (13 is prefixed or appended).
Cf. A164329 (0 is inserted), A216169 (subset of composite terms), A215417 (subset of primes), A159236 (0 is inserted between all digits).

is(n,L=logint(n+!n,10)+1,d,P)={!for(k=0,L,isprime((d=divrem(n,P=10^(L-k)))[2]+(10*d[1]+9)*P)||return)} \\ M. F. Hasler, May 10 2018

More terms from Vladeta Jovovic, Apr 16 2002

Corrected offset by Chai Wah Wu, Oct 10 2019

A217044 Primes that remain prime when a single "2" digit is inserted between any two adjacent decimal digits.

Original entry on oeis.org

17, 23, 29, 41, 53, 83, 89, 101, 113, 131, 137, 149, 251, 359, 401, 419, 443, 461, 647, 719, 797, 821, 863, 941, 1289, 1823, 2111, 2543, 3323, 3413, 4013, 4463, 4751, 5021, 5501, 5807, 6299, 6827, 7229, 7643, 7883, 8039, 8219, 8609, 8837, 9221, 9227, 9461, 9623

Offset: 1

Paolo P. Lava, Sep 25 2012

			9461 is prime and also 94621, 94261, 92461.

Bruno Berselli, Table of n, a(n) for n = 1..500 (first 123 terms from Paolo Lava)

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217045-A217047, A217062-A217065.

[p: p in PrimesInInterval(11, 10000) | forall{m: t in [1..#Intseq(p)-1] | IsPrime(m) where m is (Floor(p/10^t)*10+2)*10^t+p mod 10^t}]; // Bruno Berselli, Sep 26 2012

with(numtheory);
A217044:=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:
A217044(100000,2)

Select[Prime[Range[5,1200]],And@@PrimeQ[FromDigits/@Table[ Insert[ IntegerDigits[ #],2,i],{i,2,IntegerLength[#]}]]&] (* Harvey P. Dale, Oct 09 2012 *)

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

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

Original entry on oeis.org

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

A217062 Primes that remain prime when a single "9" digit is inserted between any two adjacent digits.

Original entry on oeis.org

11, 13, 17, 19, 23, 37, 41, 53, 59, 61, 97, 101, 107, 113, 149, 193, 197, 199, 227, 239, 263, 269, 271, 311, 331, 367, 409, 431, 443, 457, 499, 587, 617, 659, 661, 691, 727, 733, 751, 823, 863, 941, 967, 1009, 1423, 1571, 1709, 1759, 1973, 1993, 1997, 2063, 2137

Offset: 1

Paolo P. Lava, Sep 26 2012

			214883 is prime and also 2148893, 2148983, 2149883, 2194883 and 2914883.

Paolo P. Lava, Table of n, a(n) for n = 1..236

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044-A217047, A217063-A217065

with(numtheory);
A217062:=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:
A217062(1000000,9);

is(n)=my(v=concat([""], digits(n))); for(i=2, #v-1, v[1]=Str(v[1], v[i]); v[i]=9; if(i>2, v[i-1]=""); if(!isprime(eval(concat(v))), return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 26 2012

cp's OEIS Frontend

A247069 a(n) = number of distinct primes obtained when inserting 1 anywhere in A069246(n).

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A215417 Primes that remain prime when a single zero digit is inserted between any two adjacent digits.

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Maple

Mathematica

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A215419 Primes that remain prime when a single digit 3 is inserted between any two consecutive digits or as the leading or trailing digit.

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Maple

Mathematica

A068679 Numbers which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).

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Mathematica

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A164329 Numbers which yield a prime whenever a zero is inserted between any two digits.

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Mathematica

PARI

Extensions

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

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Maple

Mathematica

PARI

Extensions

A069833 Prefixing, suffixing or inserting a 9 in the number anywhere gives a prime.

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PARI

Extensions

A217044 Primes that remain prime when a single "2" digit is inserted between any two adjacent decimal digits.

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