A050719 -id:A050719 - 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

A050805 Inserting any digit between adjacent digits of prime p never yields another prime.

Original entry on oeis.org

439, 853, 1013, 1061, 1109, 1117, 1153, 1187, 1213, 1249, 1259, 1283, 1291, 1301, 1303, 1361, 1427, 1451, 1489, 1511, 1523, 1531, 1583, 1597, 1607, 1657, 1733, 1747, 1753, 1801, 1873, 1879, 1913, 1951, 2069, 2083, 2137, 2243, 2251, 2267, 2293, 2297

Offset: 1

Patrick De Geest, Oct 15 1999

			40309, 41319, 42327, 43339, 44349, 45359, 46369, 47379, 48389, and 49399 are all composite. Thus, 439, being prime, belongs to the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A050674-A050719, A050806.

Cf. A010051, A000040.

import Data.List (intersperse)
a050805 n = a050805_list !! (n-1)
a050805_list = filter ((all (== 0)) . f) a000040_list where
   f p = map (i $ show p) "0123456789"
   i ps d = a010051' (read $ intersperse d ps :: Integer)
-- Reinhard Zumkeller, May 07 2013

a[n_]:=Or@@PrimeQ[Table[FromDigits[Riffle[IntegerDigits[n],k]],{k,0,9}]]; Select[Prime[Range[5,350]],a[#]==False&] (* Jayanta Basu, May 30 2013 *)
Select[Prime[Range[400]],NoneTrue[Table[FromDigits[Riffle[ IntegerDigits[ #],d]],{d,0,9}],PrimeQ]&] (* Harvey P. Dale, Aug 04 2021 *)

Offset corrected by Reinhard Zumkeller, May 07 2013

A050806 Inserting any digit between adjacent digits of prime p produces exactly 1 new prime.

Original entry on oeis.org

101, 149, 163, 241, 269, 271, 317, 347, 367, 397, 409, 419, 443, 487, 509, 541, 587, 601, 641, 761, 787, 811, 821, 863, 907, 919, 1439, 1481, 1663, 1877, 2089, 2111, 2579, 2593, 2671, 2819, 2971, 3121, 3457, 3463, 3571, 3643, 3659, 3769, 3917, 4001

Offset: 1

Patrick De Geest, Oct 15 1999

			101 yields only one prime using digit '6' -> 1(6)0(6)1 -> prime 16061.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A050674-A050719, A050805.

Cf. A010051, A000040.

import Data.List (intersperse)
a050806 n = a050806_list !! (n-1)
a050806_list = filter ((== 1) . sum . f) a000040_list where
   f p = map (i $ show p) "0123456789"
   i ps d = a010051' (read $ intersperse d ps :: Integer)
-- Reinhard Zumkeller, May 07 2013

aQ[n_]:=Plus@@Boole[PrimeQ[Table[FromDigits[Riffle[IntegerDigits[n],k]],{k,0,9}]]]==1; Select[Prime[Range[5,555]],aQ[#]&] (* Jayanta Basu, May 30 2013 *)

Offset corrected by Reinhard Zumkeller, May 07 2013

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

A133321 Inserting any (identical) digit between adjacent digits of an odd semiprime k never yields a prime.

Original entry on oeis.org

15, 25, 35, 55, 65, 85, 95, 115, 121, 143, 145, 155, 185, 187, 205, 215, 235, 253, 265, 295, 299, 305, 335, 341, 355, 365, 393, 395, 411, 415, 437, 445, 451, 473, 485, 505, 515, 535, 545, 565, 583, 635, 655, 671, 679, 685, 695, 717, 745, 755, 781, 785, 815

Offset: 1

Jonathan Vos Post, Oct 18 2007

Odd semiprime analog of A050805. Trivially true for any digit if we substitute "even semiprime" for "odd semiprime." Trivially true for any semiprime which is a multiple of 5 (A001750). The nonmultiples of 5 in this sequence begin 121, 143, 187, 253, 299, 341.

			121 is in the sequence because 10201, 11211, 12221, 13231, 14241, 15251, 16261, 17271, 18281, 19291 are all composite.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A001750, A050674-A050719, A050805, A050806.

Select[Select[Range[11,900,2],PrimeOmega[#]==2&],AllTrue[Table[ FromDigits[ Riffle[ IntegerDigits[#],n]],{n,0,9}],CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 07 2018 *)

More terms from R. J. Mathar, Oct 22 2007

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

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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

PARI

Python

A050805 Inserting any digit between adjacent digits of prime p never yields another prime.

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Haskell

Mathematica

Extensions

A050806 Inserting any digit between adjacent digits of prime p produces exactly 1 new prime.

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