A096246 -id:A096246 - cp's OEIS frontend

A080608 Deletable primes: primes such that removing some digit leaves either the empty string or another deletable prime (possibly preceded by some zeros).

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 157, 163, 167, 173, 179, 193, 197, 223, 229, 233, 239, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439

Offset: 1

David W. Wilson, Feb 25 2003

Subsequence of A179336. - Reinhard Zumkeller, Jul 11 2010
Leading zeros are allowed in the number that appears after the digit is deleted. For example the prime 100003 is deletable because of the sequence 00003, 0003, 003, 03, 3 consists of primes. Because of this, it appears that deletable primes are relatively common in the region just above a power of ten. For example 10^1000 + 2713 is a deletable prime. - Jeppe Stig Nielsen, Aug 01 2018
For a version that does not allow leading zeros, see A305352. - Jeppe Stig Nielsen, Aug 01 2018

			410256793 is a deletable prime since each member of the sequence 410256793, 41256793, 4125673, 415673, 45673, 4567, 467, 67, 7 is prime (Weisstein, Caldwell).

David W. Wilson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deletable Prime

Cf. A080603, A096235-A096246, A305352.

read("transforms"):
isA080608 := proc(n)
    option remember;
    local dgs,i ;
    if isprime(n) then
        if n < 10 then
            true;
        else
            dgs := convert(n,base,10) ;
            for i from 1 to nops(dgs) do
                subsop(i=NULL,dgs) ;
                digcatL(ListTools[Reverse](%)) ;
                if procname(%)  then
                    return true;
                end if;
            end do:
            false ;
        end if;
    else
        false;
    end if;
end proc:
n := 1;
for i from 1 to 500 do
    p := ithprime(i) ;
    if isA080608(p) then
        printf("%d %d\n",n,p) ;
        n := n+1 ;
    fi ;
end do: # R. J. Mathar, Oct 11 2014

Rest@ Union@ Nest[Function[{a, p}, Append[a, With[{w = IntegerDigits[p]}, If[# == True, p, 0] &@ AnyTrue[Array[FromDigits@ Delete[w, #] &, Length@ w], ! FreeQ[a, #] &]]]] @@ {#, Prime[Length@ # + 1]} &, Prime@ Range@ PrimePi@ 10, 81] (* Michael De Vlieger, Aug 02 2018 *)

is(n) = !ispseudoprime(n)&&return(0);my(d=digits(n));#d==1&&return(1);for(i=1,#d,is(fromdigits(vecextract(d,Str("^"i))))&&return(1));0 \\ Jeppe Stig Nielsen, Aug 01 2018

use ntheory ":all"; sub is { my $n=shift; return 0 unless is_prime($n); my @d=todigits($n); return 1 if @d==1; is(fromdigits([vecextract(\@d,~(1<<$))])) && return 1 for 0..$#d; 0; } # _Dana Jacobsen, Nov 16 2018

from sympy import isprime
def ok(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(ok(int(t)) for t in si)
print([k for k in range(440) if ok(k)]) # Michael S. Branicky, Jan 28 2023

A080603 Primes such that deleting some digit yields a prime.

Original entry on oeis.org

13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367

Offset: 1

David W. Wilson, Feb 25 2003

Leading zeros are allowed in the number that appears after the digit is deleted, as in A080608. - Michael S. Branicky, Jan 28 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A080608, A096235-A096246.

Q@n_:=AnyTrue[FromDigits@Delete[IntegerDigits@n,#]&/@Range@IntegerLength@n, PrimeQ]; Select[Prime@Range@500, Q@# &] (* Hans Rudolf Widmer, Jun 09 2024 *)

from sympy import isprime
def ok(n):
    if n < 10 or not isprime(n): return False
    s = str(n)
    si = (s[:i]+s[i+1:] for i in range(len(s)))
    return any(isprime(int(t)) for t in si)
print([k for k in range(368) if ok(k)]) # Michael S. Branicky, Jan 28 2023

A096235 Number of n-bit base-2 deletable primes.

Original entry on oeis.org

0, 2, 2, 2, 3, 6, 6, 11, 18, 31, 49, 87, 155, 253, 427, 781, 1473, 2703, 5094, 9592, 18376, 35100, 67183, 129119, 249489, 482224, 930633, 1803598, 3502353, 6813094, 13271996, 25892906, 50583039, 98948426, 193629933, 379398057, 744508765, 1461801309

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. 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.

			d base-2 d-digit deletable primes
2 2=10, 3=11
3 5=101, 7=111
4 11=1011, 13=1101
5 19=10011, 23=10111, 29=11101
6 37=100101, 43=101011, 47=101111, 53=110101, 59=111011, 61=111101
7 73=1001001, 79=1001111, 83=1010011, 101=1100101, 107=1101011, 109=1101101

Cf. A080608, A080603, A096236-A096246.

a = {0, 2}; d = {2, 3};
For[n = 3, n <= 15, n++,
p = Select[Range[2^(n - 1), 2^n - 1], PrimeQ[#] &];
ct = 0;
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]]]; ct++;
     Break[]]]];
AppendTo[a, ct]];
a (* Robert Price, Nov 11 2018 *)

from sympy import isprime
def ok(n, prevset):
    if not isprime(n): return False
    b = bin(n)[2:]
    bi = (b[:i]+b[i+1:] for i in range(len(b)))
    return any(t[0] != '0' and int(t, 2) in prevset for t in bi)
def afind(terms):
    s, snxt = {2, 3}, set()
    print("0,", len(s), end=", ")
    for n in range(3, terms+1):
        for i in range(2**(n-1), 2**n):
            if ok(i, s):
                snxt.add(i)
        s, snxt = snxt, set()
        print(len(s), end=", ")
afind(20) # Michael S. Branicky, Jan 14 2022

a(19)-a(30) from Ryan Propper, Jul 18 2005
a(31)-a(33) from Michael S. Branicky, Jan 14 2022
a(34)-a(37) from Michael S. Branicky, May 30 2025
a(38) from Michael S. Branicky, Jun 02 2025

A321657 Base-4 deletable primes (written in base 10).

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 79, 83, 103, 107, 109, 113, 127, 139, 151, 157, 163, 167, 173, 179, 181, 191, 197, 211, 223, 227, 229, 239, 241, 251, 263, 269, 271, 283, 307, 311, 317, 331, 359, 383, 397, 419, 431, 433, 439, 443

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.

Robert Price, Table of n, a(n) for n = 1..2796

Cf. A080608, A080603, A096235-A096246.

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[]]]];
d (* Robert Price, Dec 06 2018 *)

A305352 Deletable primes (A080608) under the stricter rule that leading zeros are disallowed.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 157, 163, 167, 173, 179, 193, 197, 223, 229, 233, 239, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439

Offset: 1

Jeppe Stig Nielsen, Aug 01 2018

Subset of A080608. Numbers 2003, 2017, 2053, 3023, ... are in A080608 but not here.

If you start from a one-digit prime, you can try to build larger and larger deletable primes by inserting digits. If at one point you get stuck and cannot enlarge the number, you have reached a non-insertable prime (A125001). - Jeppe Stig Nielsen, Mar 28 2021

			2003 is not a member since removing a digit will either give 003 which has a leading zero, or give one of the numbers 203 or 200 which are both composite. However, 2003 is in A080608 because all of 2003, 003, 03, 3 are prime.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000

Cf. A080608, A080603, A096235-A096246, A125001, A322474.

Rest@ Union@ Nest[Function[{a, p}, Append[a, With[{w = IntegerDigits[p]}, If[# == True, p, 0] &@ AnyTrue[Array[If[First@ # == 0, 1, FromDigits@ #] &@ Delete[w, #] &, Length@ w], ! FreeQ[a, #] &]]]] @@ {#, Prime[Length@ # + 1]} &, Prime@ Range@ PrimePi@ 10, 81] (* Michael De Vlieger, Aug 02 2018 *)

is(n) = !ispseudoprime(n)&&return(0);my(d=digits(n));#d==1&&return(1);for(i=1,#d,my(v=vecextract(d,Str("^"i)));v[1]!=0&&is(fromdigits(v))&&return(1));0

from sympy import isprime
def ok(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 ok(int(t)) for t in si)
print([k for k in range(440) if ok(k)]) # Michael S. Branicky, Jan 28 2023

A319596 Base-3 deletable primes (written in base 10).

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 23, 29, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 137, 167, 173, 179, 181, 191, 197, 223, 233, 251, 263, 269, 317, 431, 461, 491, 503, 509, 521, 541, 547, 557, 569, 587, 593, 653, 659, 673, 677, 683, 701, 709, 719, 809, 911, 947, 953

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.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 177 terms from Robert Price)

Cf. A080608, A080603, A096235-A096246.

S:= {2}: count:= 0:
p:= 2;
while count < 200 do
  p:= nextprime(p);
  d:= floor(log[3](p));
  for i from 0 to d do
    x:= p mod 3^(i+1);
    q:= (x mod 3^i) + (p-x)/3;
    if q >= 3^(d-1) and member(q,S) then
       S:= S union {p}; count:= count+1; break
      fi
  od;
od:
sort(convert(S,list)); # Robert Israel, Nov 26 2020

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[]]]];
d (* Robert Price, Dec 05 2018 *)

from sympy import isprime
from sympy.ntheory.digits import digits
def ok(n, base=3):
    if not isprime(n): return False
    if n < 3: return True
    s = "".join(str(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 ok(int(t, base)) for t in si)
print([k for k in range(954) if ok(k)]) # Michael S. Branicky, Jan 14 2022

Removed the term 3. As pointed out by Kevin Ryde, there is no need to "seed" the list using base-2 assumptions. - Robert Price, Dec 05 2018

A322443 Base-8 deletable primes (written in base 10).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 83, 89, 101, 107, 109, 131, 137, 139, 151, 157, 163, 167, 179, 181, 191, 197, 199, 211, 223, 229, 233, 239, 251, 269, 277, 293, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 421, 431, 443, 461, 467, 479, 491

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.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..566 from Robert Price)

Cf. A080608, A080603, A096235-A096246.

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[]]]];
d (* Robert Price, Dec 08 2018 *)

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    if n < 8: return True
    o = oct(n)[2:]
    oi = (o[:i]+o[i+1:] for i in range(len(o)))
    return any(t[0] != '0' and ok(int(t, 8)) for t in oi)
print([k for k in range(492) if ok(k)]) # Michael S. Branicky, Jan 13 2022

A096236 Number of n-digit base-3 deletable primes.

Original entry on oeis.org

1, 2, 4, 7, 13, 24, 38, 72, 122, 226, 400, 684, 1246, 2381, 4384, 8330, 15839, 30617, 58764, 113987, 221994, 434498, 852036, 1673320, 3296641, 6509179

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 zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.

Cf. A080608, A080603, A096235-A096246.

b = 3; a = {1}; d = {2};
For[n = 2, n <= 10, 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 12 2018 *)

from sympy import isprime
from sympy.ntheory.digits import digits
def ok(n, prevset, base=3):
    if not isprime(n): return False
    s = "".join(str(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):
    s, snxt, base = {2}, set(), 3
    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(13) # Michael S. Branicky, Jan 14 2022

More terms from John W. Layman, Dec 14 2004

11 more terms from Ryan Propper, Jul 19 2005

A096243 Number of n-digit base-10 deletable primes.

Original entry on oeis.org

4, 16, 94, 585, 3788, 25768, 182762, 1340905, 10135727, 78580647, 622188500

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.

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, A080603, A096235-A096246.

b = 10; a = {4}; d = {2, 3, 5, 7};
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
def ok(n, prevset):
    if not isprime(n): return False
    s = str(n)
    si = (s[:i]+s[i+1:] for i in range(len(s)))
    return any(t[0] != '0' and int(t) in prevset for t in si)
def afind(terms):
    s, snxt = {2, 3, 5, 7}, set()
    print(len(s), end=", ")
    for n in range(2, terms+1):
        for i in range(10**(n-1), 10**n):
            if ok(i, s):
                snxt.add(i)
        s, snxt = snxt, set()
        print(len(s), end=", ")
afind(6) # Michael S. Branicky, Jan 14 2022

a(6)-a(9) from Ryan Propper, Jul 19 2005
a(10) from Michael S. Branicky, Jan 14 2022
a(11) from Michael S. Branicky, Jul 06 2023

A321700 Base-5 deletable primes (written in base 10).

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 37, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 103, 107, 113, 137, 197, 227, 239, 263, 269, 271, 307, 311, 317, 337, 347, 353, 359, 367, 373, 379, 389, 397, 439, 449, 479, 487, 503, 523, 547, 557, 563, 569, 571, 607, 613, 677, 727, 739, 887, 947, 977

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.

Robert Price, Table of n, a(n) for n = 1..1511

Cf. A080608, A080603, A096235-A096246.

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[]]]];
d (* Robert Price, Dec 06 2018 *)

cp's OEIS Frontend

A080608 Deletable primes: primes such that removing some digit leaves either the empty string or another deletable prime (possibly preceded by some zeros).

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A080603 Primes such that deleting some digit yields a prime.

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A096235 Number of n-bit base-2 deletable primes.

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A321657 Base-4 deletable primes (written in base 10).

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Mathematica

A305352 Deletable primes (A080608) under the stricter rule that leading zeros are disallowed.

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Mathematica

PARI

Python

A319596 Base-3 deletable primes (written in base 10).

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Maple

Mathematica

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Extensions

A322443 Base-8 deletable primes (written in base 10).

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Mathematica

Python

A096236 Number of n-digit base-3 deletable primes.