A069246 -id:A069246 - cp's OEIS frontend

A217065 Primes that remain prime when a single "7" digit is inserted between any two adjacent digits.

13, 19, 67, 73, 97, 277, 367, 379, 421, 433, 487, 541, 691, 757, 853, 967, 1117, 1471, 1747, 2017, 2617, 2749, 2851, 2953, 3463, 3529, 3571, 4507, 5077, 5923, 6073, 6079, 6343, 6481, 6577, 6709, 6829, 6967, 7351, 7417, 7573, 7681, 8317, 8719, 9157, 9649, 13177

Offset: 1

Paolo P. Lava, Sep 26 2012

			311683 is prime and also 3116873, 3116783, 3117683, 3171683 and 3711683.

Harvey P. Dale, Table of n, a(n) for n = 1..422 (First 118 terms from Paolo P. Lava)

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044-A217047, A217062-A217064.

with(numtheory);
A217065:=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:
A217065(1000000,7);

Select[Prime[Range[5,1600]],AllTrue[FromDigits/@Table[Insert[ IntegerDigits[ #],7,i],{i,2,IntegerLength[#]}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 12 2016 *)

is(n)=my(v=concat([""], digits(n))); for(i=2, #v-1, v[1]=Str(v[1], v[i]); v[i]=7; if(i>2, v[i-1]=""); if(!isprime(eval(concat(v))), return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 26 2012

A304244 Numbers that yield a prime when prime(k) is inserted after the k-th digit, for any k >= 1, k < number of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 23, 27, 29, 41, 51, 53, 77, 81, 83, 87, 89, 99, 101, 149, 191, 239, 251, 287, 317, 353, 359, 419, 473, 497, 509, 527, 533, 611, 677, 743, 797, 809, 821, 887, 893, 941, 983, 1037, 1043, 1277, 1421, 1841, 1853, 1973, 1979, 2543

Offset: 1

M. F. Hasler, May 21 2018

The primes to insert are: 2 (after the first digit), 3 (after the second digit, if there are at least three), etc.
Inspired by A304243 and analog sequences given in cross-references.
The sequence is finite: if insertion of 3 after the second digit yields a prime, then the sum of digits must be congruent to 1 or 2 (mod 3). However, insertion of 2 after the first digit also must yield a prime, so only the second case is possible. But then, insertion of a digit 7 cannot yield a prime, so no term can have 5 digits or more. (Sequence A304243 circumvents this restriction by excluding 3 from the primes to insert, but it is still finite for a similar reason occurring later.)

			The 1-digit numbers 0..9 are included since the condition is voidly satisfied: Nothing can be inserted, therefore each of the resulting numbers is prime.
17 is in the sequence because 127 is prime.
101 is in the sequence because 1201 and 1031 are prime.

M. F. Hasler, Table of n, a(n) for n = 1..100 (complete sequence).

Cf. A304243 (2 is prefixed or prime(k+2) is inserted after the k-th digit).
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).
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,P)={!for(k=1,L-1, isprime((d=divrem(n,P=10^(L-k)))[2]+(10^logint(10*p=prime(k),10)*d[1]+p)*P)|| return)}

A216165 Composite numbers and 1 which yield a prime whenever a 1 is inserted anywhere in them, including at the beginning or end.

Original entry on oeis.org

1, 49, 63, 81, 91, 99, 117, 123, 213, 231, 279, 319, 427, 459, 621, 697, 721, 801, 951, 987, 1113, 1131, 1261, 1821, 1939, 2101, 2149, 2211, 2517, 2611, 3151, 3219, 4011, 4411, 4887, 5031, 5361, 6231, 6487, 7011, 7209, 8671, 9141, 9801, 10051, 10161, 10281

Offset: 1

Paolo P. Lava, Sep 03 2012

			7209 is not prime but 72091, 72019, 72109, 71209 and 17209 are all primes.

Paolo P. Lava, Table of n, a(n) for n = 1..250

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

[n: n in [1..11000] | not IsPrime(n) and forall{m: t in [0..#Intseq(n)] | IsPrime(m) where m is (Floor(n/10^t)*10+1)*10^t+n mod 10^t}]; // Bruno Berselli, Sep 03 2012

with(numtheory);
A216165:=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:
A216165(1000,1);

Join[{1},Select[Range[11000],CompositeQ[#]&&AllTrue[FromDigits/@ Table[ Insert[ IntegerDigits[#],1,i],{i,IntegerLength[#]+1}],PrimeQ]&]] (* Harvey P. Dale, Mar 24 2017 *) (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 24 2017 *)

A304243 Numbers that yield a prime when prime(k+2) is inserted after the k-th digit (or prime(1) = 2 before the 1st digit for k=0), for 0 <= k <= number of digits.

Original entry on oeis.org

27, 33, 39, 57, 93, 333, 3747, 5073, 5997, 7239, 10053, 22419, 349731, 425991, 714807, 1719279, 81453303, 406253439, 481683189, 886662423, 2653294371

Offset: 1

M. F. Hasler, May 10 2018

The primes to insert are 2 (in front) or 5, 7, 11, 13, ... after the number's first, second, third, ... digit. So there cannot be any 1 digit solution because if 5 is appended this cannot yield a prime. One can show that the terms cannot have more than 21 digits.

The prime 3 is excluded from the strings to insert, because else no term could have more than 2 digits: to be prime with 2 prefixed or with 3 inserted, the number must be congruent to 2 (mod 3), so it cannot be prime with 7 appended or inserted. See also the Rivera link and A304244.

			a(1) = 27 because 2|27 = 227, 2|5|7 = 257 and 27|7 = 277 are all prime.
Similarly for a(6) = 333, because 2333, 3533, 3373 and 33311 are all prime.

Carlos Rivera, Puzzle 791. Interesting consecutive primes, The Prime Puzzles & Problems Connection.

Cf. A304244 (prime(k) is inserted after the k-th digit), A304245 (2 is inserted after the first digit, or prime(k+1) is inserted after the k-th digit for k > 1).
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).
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,P)={isprime(n+2*10^L) && !for(k=1,L, isprime((d=divrem(n,P=10^(L-k)))[2]+(10^logint(10*p=prime(2+k),10)*d[1]+p)*P)|| return)}

A304245 Numbers that yield a prime when '2' is inserted between the first and second digit, or prime(k+1) is inserted after the k-th digit for any k > 1, k < number of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 23, 27, 29, 41, 51, 53, 77, 81, 83, 87, 89, 99, 101, 113, 129, 149, 159, 179, 191, 203, 213, 221, 237, 251, 267, 269, 273, 281, 287, 293, 297, 321, 329, 357, 359, 401, 417, 419, 429, 441, 461, 471, 497, 509, 531, 561, 581, 603, 611, 663, 669, 687, 699, 707, 711

Offset: 1

M. F. Hasler, May 21 2018

The primes to be inserted are: 2 between 1st and 2nd digit, or 5 between 2nd and 3rd digit, or 7 between 3rd and 4th digit, etc.
The prime 3 is excluded because it would restrict the terms to have no more than 4 digits; see A304244 and the Rivera link in A304243.
The two terms 27 and 87 are the only numbers (with more than one digit) for which 2, 5 or 7 can be inserted between any two digits to yield a prime: all of 227, 257, 277, 827, 857 an 877 are prime. There is no other such number with more than 2 digits.

			The 1-digit numbers 0..9 are included since the condition is voidly satisfied: nothing can be inserted, therefore each of the resulting numbers is prime.
17 is in the sequence because 127 is prime.
101 is in the sequence because 1201 and 1051 are prime.

M. F. Hasler, Table of n, a(n) for n = 1..348 (all terms < 10^7).

Cf. A304243 (2 is prefixed or prime(k+2) is inserted after the k-th digit), A304244 (prime(k) is inserted after the k-th digit) .
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).
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,P)={!for(k=1,L-1, isprime((d=divrem(n,P=10^(L-k)))[2]+(10^logint(10*p=prime(k+(k>1)),10)*d[1]+p)*P)|| return)}

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

A217046 Primes that remain prime when a single "6" digit is inserted between any two adjacent decimal digits.

Original entry on oeis.org

13, 17, 23, 29, 37, 41, 43, 47, 53, 59, 61, 71, 79, 83, 97, 101, 109, 113, 137, 157, 163, 167, 263, 277, 293, 307, 313, 317, 331, 397, 421, 443, 457, 463, 569, 607, 653, 659, 661, 673, 691, 739, 769, 787, 809, 823, 829, 863, 881, 977, 997, 1063, 1087, 1453

Offset: 1

Paolo P. Lava, Sep 25 2012

			185917 is prime and also 1859167, 1859617, 1856917, 1865917 and 1685917.

Harvey P. Dale, Table of n, a(n) for n = 1..500 (First 262 terms from Paolo P. Lava)

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044, A217045, A217047, A217062-A217065

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(100000,6)

Select[Prime[Range[5,1200]],And@@PrimeQ[FromDigits/@Table[ Insert[ IntegerDigits[ #],6,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]=6; if(i>2, v[i-1]=""); if(!isprime(eval(concat(v))), return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 26 2012

A217063 Primes that remain prime when a single "3" digit is inserted between any two adjacent decimal digits.

Original entry on oeis.org

11, 17, 19, 23, 29, 31, 37, 41, 43, 61, 73, 79, 89, 97, 101, 103, 127, 167, 173, 181, 211, 233, 239, 251, 271, 283, 307, 331, 359, 373, 439, 491, 509, 523, 547, 599, 673, 709, 733, 769, 877, 887, 937, 941, 991, 1033, 1229, 1381, 1619, 1721, 1759, 1789, 1901

Offset: 1

Paolo P. Lava, Sep 26 2012

			212881 is prime and also 2128831, 2128381, 2123881, 213288 and 2312881.

Bruno Berselli, Table of n, a(n) for n = 1..1000 (first 275 terms from Paolo Lava)

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

[p: p in PrimesInInterval(11, 2000) | forall{m: t in [1..#Intseq(p)-1] | IsPrime(m) where m is (Floor(p/10^t)*10+3)*10^t+p mod 10^t}]; // Bruno Berselli, Sep 26 2012

with(numtheory);
A217063:=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:
A217063(1000000,3);

is(n)=my(v=concat([""], digits(n))); for(i=2, #v-1, v[1]=Str(v[1], v[i]); v[i]=3; if(i>2, v[i-1]=""); if(!isprime(eval(concat(v))), return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 26 2012

from sympy import isprime, primerange
def ok(p):
    if p < 10: return False
    s = str(p)
    return all(isprime(int(s[:i] + "3" + 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(1901)) # Michael S. Branicky, Nov 17 2021

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

Original entry on oeis.org

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

Paolo P. Lava, Sep 03 2012

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

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

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

[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

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);

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 *)

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

A217065 Primes that remain prime when a single "7" digit is inserted between any two adjacent digits.

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A304244 Numbers that yield a prime when prime(k) is inserted after the k-th digit, for any k >= 1, k < number of digits.

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A216165 Composite numbers and 1 which yield a prime whenever a 1 is inserted anywhere in them, including at the beginning or end.

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Magma

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Mathematica

A304243 Numbers that yield a prime when prime(k+2) is inserted after the k-th digit (or prime(1) = 2 before the 1st digit for k=0), for 0 <= k <= number of digits.

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A304245 Numbers that yield a prime when '2' is inserted between the first and second digit, or prime(k+1) is inserted after the k-th digit for any k > 1, k < number of 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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Maple

Mathematica

PARI

A217046 Primes that remain prime when a single "6" digit is inserted between any two adjacent decimal digits.

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Maple

Mathematica

PARI

A217063 Primes that remain prime when a single "3" digit is inserted between any two adjacent decimal digits.

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