A215420 -id:A215420 - cp's OEIS frontend

A215421 Primes that remain prime when a single digit 9 is inserted between any two consecutive digits or as the leading or trailing digit.

7, 19, 37, 41, 199, 239, 311, 587, 661, 941, 967, 1009, 1997, 4993, 4999, 5393, 5651, 6911, 9109, 9397, 9679, 9829, 19417, 20233, 22549, 27397, 29389, 31387, 39989, 71419, 71569, 90599, 91951, 95369, 97103, 98909, 99023, 160009, 225919, 267389, 313991, 328849

Offset: 1

Paolo P. Lava, Aug 10 2012

			31387 is prime and also 313879, 313897, 313987, 319387, 391387, 931387.

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A215417-A215420

A215421:=proc(q,x)
local a,b,c,d,i,n,ok;
for n from 1 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 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(ithprime(n)); fi;
od; end:
A215421(1000,9);

Select[Prime[Range[30000]],AllTrue[FromDigits/@Table[Insert[ IntegerDigits[ #],9,n],{n, IntegerLength[ #]+1}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 22 2016 *)

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 *)

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

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

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)}

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)}

A304246 Numbers that yield a prime whenever a '1' is inserted between any two digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 21, 31, 33, 37, 49, 63, 67, 69, 79, 81, 91, 99, 103, 109, 117, 123, 151, 163, 181, 193, 211, 213, 231, 241, 279, 309, 319, 363, 367, 391, 411, 427, 429, 453, 457, 459, 501, 513, 519, 547, 571, 601, 613, 621, 631, 697, 703, 709, 721, 729, 777, 787, 801, 811, 817, 879, 903, 951, 981, 987

Offset: 1

M. F. Hasler, Jun 01 2018

The single-digit terms voidly satisfy the condition: no '1' can be inserted anywhere, so all possible insertions yield a prime.
Motivated by sequence A164329 which is the analog for inserting 0.
Compare to A068673 where 1 is prefixed or appended, and to A068679 where 1 is prefixed, appended or inserted anywhere - which is therefore the intersection between this sequence and A068673.
See also A050711 where 1 is inserted between all adjacent digits. - R. J. Mathar, Feb 28 2020

			21 is in the sequence, because if '1' is inserted between "any" pair consecutive digits (the only possibility being to insert it between the first and second digit, which yields 211), the result is always prime. The definition does not require the term itself to be prime.
103 is in the sequence because inserting 1 between the first and second, or between the second and third digit, would yield 1103 or 1013, respectively, which are both prime.

Cf. A164329 (prime when 0 is inserted anywhere), A216169 (subset of composite terms), A215417 (subset of primes), A159236 (prime when 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).
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), A304245 (prime(k+1) is inserted after the k-th digit, k > 1, or '2' after the first digit).

[0] cat [k:k in [1..1000]| forall{i:i in [1..#Intseq(k)-1]| IsPrime(Seqint(Reverse(v[1..i] cat [1] cat v[i+1..#v]))) where v is Reverse(Intseq(k)) }]; // Marius A. Burtea, Feb 09 2020

filter:= proc(n) local j,t;
  for j from 1 to ilog10(n) do
    if not isprime(10*n-9*(n mod 10^j)+10^j) then return false fi
  od;
  true
end proc:
select(filter, [$0..1000]); # Robert Israel, Jun 01 2018

is(n)=!for(k=1,logint(n+!n,10),isprime(10*n-n%10^k*9+10^k)||return)

A304247 Numbers which yield a prime whenever a '2' is inserted between any single pair of adjacent 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, 123, 129, 131, 137, 149, 183, 207, 221, 243, 251, 297, 303, 321, 329, 357, 359, 399, 401, 417, 419, 429, 441, 443, 453, 461, 471, 473, 527, 533, 581, 597, 611, 621

Offset: 1

M. F. Hasler, Jun 01 2018

Motivated by existing sequences defined in an analog way for other digits to be inserted, e.g., A164329 for the digit 0, cf. cross-references.
For single-digit terms, the condition is voidly satisfied: nothing can be inserted.
See also A050712 where 2 is inserted between each pair of adjacent digits. - R. J. Mathar, Feb 28 2020

			123 is in the sequence because it yields a prime when a '2' is inserted after the first or after the second digit, which yields the prime 1223 in both cases. The term itself does not need to be prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A164329 (prime when 0 is inserted anywhere), 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), A304246 (1 is inserted anywhere).
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. 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), A304245 (prime(k+1) is inserted after the k-th digit, k > 1, or '2' after the first digit).

filter:= proc(n) local j,t;
  for j from 1 to ilog10(n) do
    if not isprime(10*n-9*(n mod 10^j)+2*10^j) then return false fi
  od;
  true
end proc:
select(filter, [$0..10000]); # Robert Israel, Jun 01 2018

is(n,p=2,L=logint(n+!n,10)+1,d,P)=!for(k=1,L-1,isprime((d=divrem(n,P=10^(L-k)))[2]+(10*d[1]+p)*P)||return)

A228149 a(n) is the smallest n-digit prime which remains prime whenever a 7 is inserted, appended or prepended, or 0 if no such prime exists.

Original entry on oeis.org

3, 19, 433, 1471, 13177, 175687, 1740787, 10871017, 137725897, 1670181787, 12845952529, 117777772309, 1008078796477, 11777787780727

Offset: 1

Giovanni Resta, Aug 13 2013

Smallest n-digit term of A215420.

			a(3) = 433 is the smallest 3-digit prime which yields 4 primes by inserting a 7 in every possible position: 7433, 4733, 4373, 4337.

Cf. A215420, A164328, A228148, A228150.

pp[n_, k_] := Catch[Block[{d = IntegerDigits@n}, Do[If[! PrimeQ[ FromDigits[ Insert[d, k, i]]], Throw[False]], {i, 1+Length@d}]; True]]; a[n_] := Catch[ Block[{p = NextPrime[10^(n-1)]}, While[p < 10^n, If[pp[p, 7], Throw@p, p = NextPrime@p]]; 0]]; a /@ Range[8]

cp's OEIS Frontend

A215421 Primes that remain prime when a single digit 9 is inserted between any two consecutive digits or as the leading or trailing digit.

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Maple

Mathematica

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

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

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PARI

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

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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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PARI

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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PARI

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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PARI

A304246 Numbers that yield a prime whenever a '1' is inserted between any two digits.

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