A068679 -id:A068679 - cp's OEIS frontend

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

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

A216168 Composite numbers and 1 which yield a prime whenever a 7 is inserted anywhere in them, including at the beginning or end.

Original entry on oeis.org

1, 9, 27, 33, 39, 57, 87, 159, 177, 187, 603, 717, 753, 949, 1257, 1707, 2277, 2367, 4317, 4623, 4779, 4797, 5773, 6757, 6777, 7017, 7471, 7479, 7747, 7797, 7813, 7977, 8797, 9777, 9987, 10777, 11757, 17679, 28269, 28437, 29779, 34177, 34771, 40059, 41721

Offset: 1

Paolo P. Lava, Sep 03 2012

			4623 is not prime but 46237, 46273, 46723, 47623 and 74623 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+7)*10^t+n mod 10^t}]; // Bruno Berselli, Sep 03 2012

with(numtheory);
A216168:=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:
A216168(1000,7);

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)

A304248 Numbers that yield a prime whenever a '3' is inserted between any two digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 17, 19, 23, 29, 31, 37, 41, 43, 49, 61, 73, 79, 89, 97, 101, 103, 121, 127, 167, 173, 181, 209, 211, 233, 239, 247, 251, 271, 283, 299, 307, 331, 343, 359, 361, 373, 391, 437, 439, 473, 491, 497, 509, 523, 533, 547, 551, 599

Offset: 1

M. F. Hasler, Jun 01 2018

Motivated by existing sequences defined in a similar way for other digits (e.g., A164329 for digit 0), subsequence A158594 = intersection of this and A068674 ('3' is prefixed or appended), and others: cf. cross-references.

			121 is in the sequence because it yields a prime when a digit 3 is inserted after the first or after the second digit, which yields the prime 1321 or 1231, respectively. The term itself does not need to be prime.
The single-digit numbers 0..9 are in the sequence because they satisfy the condition voidly: nothing can be inserted, so no insertion yields a nonprime, so all possible insertions always yield a 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 (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), A304246 (1 is inserted anywhere).
Cf. A304247 (2 is inserted anywhere).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these), A068674 (3 is prefixed or appended).
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..600]| forall{i:i in [1..#Intseq(k)-1]| IsPrime(Seqint(Reverse(v[1..i] cat [3] cat v[i+1..#v]))) where v is Reverse(Intseq(k))}]; // Marius A. Burtea, Feb 09 2020

Select[Range[0,600],AllTrue[FromDigits/@Table[Insert[IntegerDigits[#],3,n],{n,2,IntegerLength[ #]}],PrimeQ]&] (* Harvey P. Dale, Nov 06 2022 *)

is(n, p=3, 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)

cp's OEIS Frontend

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

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Magma

Maple

Mathematica

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

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Magma

Maple

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Python

A216168 Composite numbers and 1 which yield a prime whenever a 7 is inserted anywhere in them, including at the beginning or end.

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Magma

Maple

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

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Magma

Maple

PARI

A304247 Numbers which yield a prime whenever a '2' is inserted between any single pair of adjacent digits.

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Comments

Examples

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Crossrefs

Programs

Maple

PARI

A304248 Numbers that yield a prime whenever a '3' is inserted between any two digits.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI