A050711 -id:A050711 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A050715 Inserting a digit '5' between adjacent digits of n makes a prime.

Original entry on oeis.org

11, 17, 21, 27, 33, 39, 47, 57, 63, 69, 71, 77, 83, 87, 89, 93, 103, 129, 139, 141, 151, 159, 189, 199, 207, 213, 223, 237, 243, 247, 267, 279, 291, 301, 303, 309, 313, 319, 321, 327, 333, 373, 379, 381, 391, 403, 429, 453, 457, 469, 471, 477, 483, 493, 499

Offset: 1

Patrick De Geest, Aug 15 1999

			373 becomes 3(5)7(5)3 which is prime 35753.

Harvey P. Dale, Table of n, a(n) for n = 1..1001

Cf. A050674, A050711, A050712, A050713, A050714, A050716, A050717, A050718, A050719.

Select[Range[10,500],PrimeQ[FromDigits[Riffle[IntegerDigits[#],5]]]&] (* Harvey P. Dale, Apr 07 2018 *)

Offset changed to 1 by Georg Fischer, Oct 15 2019

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

A217064 Primes that remain prime when a single "5" digit is inserted between any two adjacent decimal digits.

Original entry on oeis.org

11, 17, 47, 71, 83, 89, 149, 167, 179, 251, 257, 293, 347, 359, 383, 419, 461, 467, 491, 557, 563, 569, 653, 773, 911, 1193, 1217, 1277, 1451, 1559, 1667, 1823, 1901, 2243, 2309, 2357, 2579, 2657, 2999, 3527, 3533, 4289, 5051, 5351, 5501, 5843, 6089, 6551, 6581

Offset: 1

Paolo P. Lava, Sep 26 2012

			290183 is prime and also 2901853, 2901583, 2905183, 2950183 and 2590183.

Harvey P. Dale, Table of n, a(n) for n = 1..500 (First 140 terms from Paolo P. Lava)

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044-A217047, A217062, A217063, A217065

with(numtheory);
A217064:=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:
A217064(1000000,5);

Select[Prime[Range[5,1000]],AllTrue[FromDigits/@Table[ Insert[ IntegerDigits[ #],5,n],{n,2,IntegerLength[#]}],PrimeQ]&] (* Harvey P. Dale, Feb 20 2022 *)

is(n)=my(v=concat([""], digits(n))); for(i=2, #v-1, v[1]=Str(v[1], v[i]); v[i]=5; if(i>2, v[i-1]=""); if(!isprime(eval(concat(v))), return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 26 2012

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)

A331116 Inserting a digit '1' between the first two adjacent digits of k, then inserting a digit '2' between the two following adjacent digits of k, ..., then inserting the integer '10' between the tenth and the eleventh digits of k, ... produces a prime number.

Original entry on oeis.org

13, 21, 31, 33, 37, 49, 63, 67, 69, 79, 81, 91, 99, 107, 131, 139, 143, 157, 161, 181, 187, 193, 197, 203, 211, 221, 227, 233, 251, 253, 259, 277, 281, 299, 311, 313, 323, 331, 337, 367, 371, 373, 377, 379, 403, 421, 427, 451, 461, 467, 479

Offset: 1

Bernard Schott, Jan 10 2020

Inspired by the sequences A050711 to A050719, so the first 13 terms are the first 13 terms of A050711, then a(14) = 107 because 1(1)0(2)7 gives 11027 which is a prime.

			281 gives 2(1)8(2)1 = 21821 that is prime, hence 281 is a term.
1027 gives 1(1)0(2)2(3)7 = 1102237 that is prime, hence 1027 is another term.

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

Cf. A050711, A050712, A050713, A050714.

Cf. A050715, A050716, A050717, A050718, A050719.

seqQ[n_] := PrimeQ @ FromDigits @ Flatten @ IntegerDigits @ Riffle[(d = IntegerDigits[n]), Range[Length[d] - 1]]; Select[Range[10,480], seqQ] (* Amiram Eldar, Jan 10 2020 *)

from sympy import isprime
def ok(n):
    if n < 10: return False
    s = str(n)
    shuffle = list(map(str, ((i+1)//2 for i in range(2*len(s)-1))))
    shuffle[0::2] = [s[i] for i in range(len(s))]
    return isprime(int("".join(shuffle)))
print(list(filter(ok, range(480)))) # Michael S. Branicky, Jul 18 2021

cp's OEIS Frontend

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

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Maple

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

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Maple

Mathematica

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A050715 Inserting a digit '5' between adjacent digits of n makes a prime.

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Mathematica

Extensions

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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Magma

Maple

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Python

A217064 Primes that remain prime when a single "5" digit is inserted between any two adjacent decimal digits.

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Keywords

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Programs

Maple

Mathematica

PARI

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

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