A080608 -id:A080608 - cp's OEIS frontend

A096245 Number of n-digit base-12 deletable primes.

5, 25, 186, 1398, 11500, 99074, 893062, 8352961, 80564801

Offset: 1

Michael Kleber, Feb 28 2003

A prime p is a base-b deletable prime if, when written in base b, it has the property that removing some digit leaves either the empty string or another deletable prime. "Digit" means digit in base b.

Deleting a digit cannot leave any leading 0's in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.

Cf. A080608, A080603, A096235-A096246.

b = 12; a = {5}; d = {2, 3, 5, 7, 11};
For[n = 2, n <= 5, n++,
  p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];
  ct = 0;
  For[i = 1, i <= Length[p], i++,
   c = IntegerDigits[p[[i]], b];
   For[j = 1, j <= n, j++,
    t = Delete[c, j];
    If[t[[1]] == 0, Continue[]];
    If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; ct++;
     Break[]]]];
  AppendTo[a, ct]];
a (* Robert Price, Nov 13 2018 *)

from sympy import isprime
from sympy.ntheory.digits import digits
def strmap(d):
    return str(d) if d < 10 else "ABCDEFGHIJKLMNOPQRSTUVWXYZ"[d-10]
def ok(n, prevset, base=12): # works for bases 2-36
    if not isprime(n): return False
    s = "".join(strmap(d) for d in digits(n, base)[1:])
    si = (s[:i]+s[i+1:] for i in range(len(s)))
    return any(t[0] != '0' and int(t, base) in prevset for t in si)
def afind(terms, base=12): # works for bases 3-36
    s = set([p for p in range(1, base) if isprime(p)])
    alst, snxt = [len(s)], set()
    print(len(s), end=", ")
    for n in range(2, terms+1):
        for i in range(base**(n-1), base**n):
            if ok(i, s):
                snxt.add(i)
        s, snxt = snxt, set()
        print(len(s), end=", ")
afind(6) # Michael S. Branicky, Jan 17 2022

a(6)-a(8) from Ryan Propper, Jul 19 2005
Edited by Charles R Greathouse IV, Aug 03 2010
a(9) from Michael S. Branicky, Jan 17 2022

A125589 Smallest n-digit base-10 deletable prime.

Original entry on oeis.org

2, 13, 103, 1013, 10039, 100103, 1000193, 10000931, 100001903, 1000003957, 10000003957, 100000013957, 1000000030957, 10000000301957, 100000000730957, 1000000000730957, 10000000003632979, 100000000007309357

Offset: 1

N. J. A. Sloane, Jan 07 2007

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime. "Digit" means digit in base b.

Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.

Cf. A080608, A096243, A096246, A125590.

b = 10; a = {2}; d = {2, 3, 5, 7};
For[n = 2, n <= 6, n++,
  found = False;
  p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];
  For[i = 1, i <= Length[p], i++,
   c = IntegerDigits[p[[i]], b];
   For[j = 1, j <= n, j++,
    t = Delete[c, j];
    If[t[[1]] == 0, Continue[]];
    If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]];
     If[! found , AppendTo[a, p[[i]]]]; found = True; Break[]]];
]]; a (* Robert Price, Nov 13 2018 *)

a(6) - a(8) from Michael Kleber, Jan 08 2007
a(9) - a(14) from Phil Carmody, Jan 09 2007
a(15) - a(18) from Joshua Zucker, Jan 09 2007

A125590 Largest n-digit base-10 deletable prime.

Original entry on oeis.org

7, 97, 997, 9973, 99929, 999907, 9999907, 99999307, 999996671, 9999996073, 99999966307, 999999908773, 9999999710639, 99999999697769, 999999997160639, 9999999996977699, 99999999980803477, 999999999961861807, 9999999999961861807, 99999999999807429133

Offset: 1

N. J. A. Sloane, Jan 07 2007

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime. "Digit" means digit in base b.

Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.

			99929 -> 9929 -> 929 -> 29 -> 2.

C. Caldwell, Truncatable primes, J. Recreational Math., 19:1 (1987) 30-33. [Discusses left truncatable primes, right truncatable primes and deletable primes.]

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
C. Caldwell, Deletable primes
Prime Curios, A 300-digit example
Carlos Rivera, Puzzle 138: Deletable Primes, Prime Puzzles and Problems Connection. [Includes a 500-digit example]
Index entries for sequences related to truncatable primes

Cf. A080608, A096243, A096246, A125589.

b = 10; a = {7}; d = {2, 3, 5, 7};
For[n = 2, n <= 5, n++,
  p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];
  For[i = 1, i <= Length[p], i++,
   c = IntegerDigits[p[[i]], b];
   For[j = 1, j <= n, j++,
    t = Delete[c, j];
    If[t[[1]] == 0, Continue[]];
    If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]];
  AppendTo[a, Last[d]]];
a (* Robert Price, Nov 13 2018 *)

from sympy import isprime, prevprime
from functools import cache
@cache
def deletable_prime(n):
    if not isprime(n): return False
    if n < 10: return True
    s = str(n)
    si = (s[:i]+s[i+1:] for i in range(len(s)))
    return any(t[0] != '0' and deletable_prime(int(t)) for t in si)
def a(n):
    p = prevprime(10**n)
    while not deletable_prime(p): p = prevprime(p)
    return p
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Jan 13 2022

a(6)-a(8) from Michael Kleber, Jan 08 2007
a(9)-a(16) from Joshua Zucker, May 11 2007
a(17)-a(20) from Michael S. Branicky, Jan 13 2022

A320587 Primes that are not Base-3 deletable primes (written in base 10).

Original entry on oeis.org

3, 13, 31, 37, 41, 43, 67, 79, 97, 103, 109, 113, 127, 131, 139, 149, 151, 157, 163, 193, 199, 211, 227, 229, 239, 241, 257, 271, 277, 281, 283, 293, 307, 311, 313, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 433, 439, 443

Offset: 1

Robert Price, Nov 14 2018

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
Complement of all primes and A319596.

Robert Price, Table of n, a(n) for n = 1..906

Cf. A080608, A080603, A096235-A096246.

b = 3; d = {};
p = Select[Range[2, 10000], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
  c = IntegerDigits[p[[i]], b];
  If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];
  For[j = 1, j <= Length[c], j++,
   t = Delete[c, j];
   If[t[[1]] == 0, Continue[]];
If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]]; Complement[Table[Prime[n], {n, PrimePi[Last[d]]}], d] (* Robert Price, Dec 06 2018 *)

Added the term 3. As pointed out by Kevin Ryde, there is no need to "seed" the list using base-2 assumptions. - Robert Price, Dec 06 2018

A321596 Primes that are not base-2 deletable primes (written in base 10).

Original entry on oeis.org

17, 31, 41, 67, 71, 89, 97, 103, 113, 127, 131, 139, 181, 191, 193, 199, 223, 227, 233, 239, 241, 251, 257, 263, 269, 271, 283, 337, 353, 367, 373, 379, 383, 401, 409, 431, 433, 439, 443, 449, 463, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 571, 577

Offset: 1

Robert Price, Nov 14 2018

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime. However, in base 2 we adopt the convention that 2 = 10 and 3 = 11 are deletable.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
Complement of all primes and A096246.

Robert Price, Table of n, a(n) for n = 1..2351

Cf. A080608, A080603, A096235-A096246.

d = {2, 3};
For[n = 3, n <= 15, n++,
  p = Select[Range[2^(n - 1), 2^n - 1], PrimeQ[#] &];
  For[i = 1, i <= Length[p], i++,
   c = IntegerDigits[p[[i]], 2];
   For[j = 1, j <= n, j++,
    t = Delete[c, j];
    If[t[[1]] == 0, Continue[]];
    If[MemberQ[d, FromDigits[t, 2]], AppendTo[d, p[[i]]];  Break[]]]]]; Complement[Table[Prime[n], {n, PrimePi[Last[d]]}], d]

A321658 Primes that are not base-4 deletable primes (written in base 10).

Original entry on oeis.org

5, 17, 37, 41, 73, 89, 97, 101, 131, 137, 149, 193, 199, 233, 257, 277, 281, 293, 313, 337, 347, 349, 353, 367, 373, 379, 389, 401, 409, 421, 521, 569, 577, 593, 601, 613, 617, 641, 661, 673, 677, 683, 761, 769, 809, 811, 823, 829, 853, 857, 859, 929, 937

Offset: 1

Robert Price, Nov 15 2018

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
Complement of all primes and A319596.

Robert Price, Table of n, a(n) for n = 1..3741

Cf. A080608, A080603, A096235-A096246, A321657.

b = 4; d = {};
p = Select[Range[2, 10000], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
  c = IntegerDigits[p[[i]], b];
  If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];
  For[j = 1, j <= Length[c], j++,
   t = Delete[c, j];
   If[t[[1]] == 0, Continue[]];
If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]]; Complement[Table[Prime[n], {n, PrimePi[Last[d]]}], d] (* Robert Price, Dec 06 2018 *)

A321701 Primes that are not base-5 deletable primes (written in base 10).

Original entry on oeis.org

5, 29, 31, 41, 43, 83, 101, 109, 127, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 199, 211, 223, 229, 233, 241, 251, 257, 277, 281, 283, 293, 313, 331, 349, 383, 401, 409, 419, 421, 431, 433, 443, 457, 461, 463, 467, 491, 499, 509, 521, 541, 577, 587, 593, 599

Offset: 1

Robert Price, Nov 17 2018

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
Complement of all primes and A321700.

Robert Price, Table of n, a(n) for n = 1..31550

Cf. A080608, A080603, A096235-A096246, A321657.

b = 5; d = {};
p = Select[Range[2, 10000], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
  c = IntegerDigits[p[[i]], b];
  If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];
  For[j = 1, j <= Length[c], j++,
   t = Delete[c, j];
   If[t[[1]] == 0, Continue[]];
If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]]; Complement[Table[Prime[n], {n, PrimePi[Last[d]]}], d] (* Robert Price, Dec 06 2018 *)

A322172 Primes that are not base-7 deletable primes (written in base 10).

Original entry on oeis.org

7, 11, 13, 29, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 151, 157, 173, 179, 181, 193, 197, 211, 229, 239, 257, 269, 271, 277, 281, 283, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499

Offset: 1

Robert Price, Nov 29 2018

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
Complement of all primes and A321910.

Robert Price, Table of n, a(n) for n = 1..10286

Cf. A080608, A080603, A096235-A096246, A321657.

b = 7; d = {};
p = Select[Range[2, 10000], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
  c = IntegerDigits[p[[i]], b];
  If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];
  For[j = 1, j <= Length[c], j++,
   t = Delete[c, j];
   If[t[[1]] == 0, Continue[]];
If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]]; Complement[Table[Prime[n], {n, PrimePi[Last[d]]}], d] (* Robert Price, Dec 06 2018 *)

A322174 Primes that are not base-6 deletable primes (written in base 10).

Original entry on oeis.org

7, 37, 43, 61, 151, 223, 229, 241, 271, 277, 307, 331, 337, 349, 367, 523, 691, 709, 733, 863, 907, 1009, 1033, 1051, 1069, 1103, 1109, 1123, 1223, 1231, 1249, 1283, 1289, 1297, 1301, 1303, 1321, 1327, 1381, 1423, 1429, 1447, 1471, 1483, 1531, 1549, 1567, 1597, 1621, 1627, 1657, 1663, 1741

Offset: 1

Robert Price, Nov 29 2018

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
Complement of all primes and A322173.

Robert Price, Table of n, a(n) for n = 1..1258

Cf. A080608, A080603, A096235-A096246, A321657.

b = 6; d = {};
p = Select[Range[2, 10000], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
  c = IntegerDigits[p[[i]], b];
  If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];
  For[j = 1, j <= Length[c], j++,
   t = Delete[c, j];
   If[t[[1]] == 0, Continue[]];
If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]]; Complement[Table[Prime[n], {n, PrimePi[Last[d]]}], d] (* Robert Price, Dec 06 2018 *)

A322444 Primes that are not base-8 deletable primes (written in base 10).

Original entry on oeis.org

71, 73, 79, 97, 103, 113, 127, 149, 173, 193, 227, 241, 257, 263, 271, 281, 283, 307, 311, 313, 409, 419, 433, 439, 449, 457, 463, 487, 503, 521, 569, 577, 587, 593, 599, 607, 617, 631, 641, 647, 653, 661, 673, 701, 727, 733, 739, 743, 757, 769, 823, 827, 839, 881, 887, 907, 911, 919, 929

Offset: 1

Robert Price, Dec 08 2018

A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
Complement of all primes and A322443 .

Robert Price, Table of n, a(n) for n = 1..660

Cf. A080608, A080603, A096235-A096246, A321657.

b = 8; d = {};
p = Select[Range[2, 10000], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
c = IntegerDigits[p[[i]], b];
If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];
For[j = 1, j <= Length[c], j++,
t = Delete[c, j];
If[t[[1]] == 0, Continue[]];
If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]]; Complement[Table[Prime[n], {n, PrimePi[Last[d]]}], d] (* Robert Price, Dec 08 2018 *)

cp's OEIS Frontend

A096245 Number of n-digit base-12 deletable primes.

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A125589 Smallest n-digit base-10 deletable prime.

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A125590 Largest n-digit base-10 deletable prime.

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A320587 Primes that are not Base-3 deletable primes (written in base 10).

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A321596 Primes that are not base-2 deletable primes (written in base 10).

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A321658 Primes that are not base-4 deletable primes (written in base 10).

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A321701 Primes that are not base-5 deletable primes (written in base 10).

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A322172 Primes that are not base-7 deletable primes (written in base 10).

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A322174 Primes that are not base-6 deletable primes (written in base 10).