A034302 -id:A034302 - cp's OEIS frontend

A051362 Primes remaining prime if any digit is deleted (zeros allowed).

23, 37, 53, 73, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1013, 1031, 1097, 1499, 1997, 2239, 2293, 3137, 4019, 4919, 6173, 7019, 7433, 9677, 10193, 10613, 11093, 19973, 23833, 26833, 30011, 37019, 40013, 47933, 73331, 74177

Offset: 1

Harvey P. Dale, May 31 2000

These might be called "super-prime numbers". - Jaime Gutierrez (jgutierrez(AT)matematicas.net), Sep 27 2007
A proper subset of A034895. - Robert G. Wilson v, Oct 12 2014
The largest known number in this sequence is a 274-digit prime consisting of 163 4s, followed by 80 0s, followed by 31 1s. See the CodeGolf link. - Dmitry Kamenetsky, Feb 26 2021

Giovanni Resta, Table of n, a(n) for n = 1..201 (terms < 10^13; first 100 terms from T. D. Noe)
CodeGolf StackExchange, Find largest prime which is still a prime after digit deletion, 2013.
Mathematics StackExchange, Deleting any digit yields a prime, 2011.
Mathematics StackExchange, Largest prime that remains prime when any one of its digits is deleted, 2021.

Cf. A034302, A010051, A000040, A034895.

import Data.List (inits, tails)
a051362 n = a051362_list !! (n-1)
a051362_list = filter p $ drop 4 a000040_list where
   p x = all (== 1) $ map (a010051 . read) $
             zipWith (++) (inits $ show x) (tail $ tails $ show x)
-- Reinhard Zumkeller, Dec 17 2011, Aug 24 2011

rpQ[n_]:=Module[{idn=IntegerDigits[n]},And@@PrimeQ[FromDigits/@ Subsets[ IntegerDigits[ n],{Length[idn]-1}]]]; Select[Prime[Range[40000]], rpQ]
prpQ[n_]:=AllTrue[FromDigits/@Table[Delete[IntegerDigits[n],d],{d,IntegerLength[ n]}],PrimeQ]; Select[Prime[Range[7500]],prpQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 27 2020 *)

is(n)=my(v=Vec(Str(n)),k);for(i=1, #v, k=eval(concat(vecextract(v, 2^#v-1-2^(i-1))));if(!isprime(k),return(0)));isprime(n) \\ Charles R Greathouse IV, Oct 05 2011

from sympy import isprime
def ok(n):
    if n < 10 or not isprime(n): return False
    s = str(n)
    return all(isprime(int(s[:i]+s[i+1:])) for i in range(len(s)))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 02 2023

def is_A051362(n):
    prime = is_prime(n)
    if prime:
        L = ZZ(n).digits(10)
        for k in range(len(L)):
            K = L[:]; del K[k]
            prime = is_prime(ZZ(K, base=10))
            if not prime: break
    return prime
A051362_list = lambda n: filter(is_A051362, range(n))
A051362_list(77777) # Peter Luschny, Jul 17 2014

A034305 Zeroless nonprimes that remain nonprime if any digit is deleted.

Original entry on oeis.org

14, 16, 18, 44, 46, 48, 49, 64, 66, 68, 69, 81, 84, 86, 88, 91, 94, 96, 98, 99, 122, 124, 125, 126, 128, 142, 144, 145, 146, 148, 152, 154, 155, 156, 158, 162, 164, 165, 166, 168, 182, 184, 185, 186, 188, 212, 214, 215, 216, 218, 221, 222, 224, 225, 226, 228

Offset: 1

David W. Wilson

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Subsequence of A052382.

Cf. A034302, A034303, A034304, A010051.

a034305 n = a034305_list !! (n-1)
a034305_list = filter f $ drop 9 a052382_list where
  f x = a010051' x == 0 &&
        (all (== 0) $ map (a010051' . read) $
         zipWith (++) (inits $ show x) (tail $ tails $ show x))
-- Reinhard Zumkeller, May 10 2015

npQ[n_]:=!PrimeQ[n]&&FreeQ[IntegerDigits[n],0]&&AllTrue[FromDigits/@ Table[Drop[IntegerDigits[n],{k}],{k,IntegerLength[n]}],!PrimeQ[#]&]; Select[Range[10,300],npQ](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 19 2021 *)

is(n)=my(d=digits(n)); if(#d<2 || vecmin(d)<1 || isprime(n), return(0)); for(i=0,#d-1, if(isprime(fromdigits(vecextract(d,2^#d-1-2^i))), return(0))); 1 \\ Charles R Greathouse IV, Jun 25 2017

from sympy import isprime
def ok(n):
    if n < 10 or isprime(n): return False
    s = str(n)
    return "0" not in s and not any(isprime(int(s[:i]+s[i+1:])) for i in range(len(s)))
print([k for k in range(229) if ok(k)]) # Michael S. Branicky, Jan 15 2023

Definition corrected by T. D. Noe, Apr 02 2008

Single-digit terms removed again by Georg Fischer, Jun 21 2021

A057876 Primes p with the following property: let d_1, d_2, ... be the distinct digits occurring in the decimal expansion of p. Then for each d_i, dropping all the digits d_i from p produces a prime number. Leading 0's are not allowed.

Original entry on oeis.org

23, 37, 53, 73, 113, 131, 137, 151, 173, 179, 197, 211, 311, 317, 431, 617, 719, 1531, 1831, 1997, 2113, 2131, 2237, 2273, 2297, 2311, 2797, 3137, 3371, 4337, 4373, 4733, 4919, 6173, 7297, 7331, 7573, 7873, 8191, 8311, 8831, 8837, 12239, 16673, 19531

Offset: 1

Patrick De Geest, Oct 15 2000

			1531 gives primes 53, 131 and 151 after dropping digits 1, 5 and 3.
A larger example 1210778071 gives primes 12177871, 2077807, 110778071, 1210801 and 121077071 after dropping digits 0, 1, 2, 7 and 8.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A057877-A057883, A051362, A034302-A034305.

filter:= proc(n) local L,d,Lp;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  for d in convert(L,set) do
    Lp:= subs(d=NULL,L);
    if Lp=[] or Lp[-1] = 0 then return false fi;
    if not isprime(add(Lp[i]*10^(i-1),i=1..nops(Lp))) then return false fi;
  od;
  true
end proc:
select(filter, [seq(i,i=13..20000,2)]); # Robert Israel, Jul 13 2018

Name edited by Robert Israel, Jul 13 2018

A057883 Smallest possible prime with at least n (from 2 to 10) distinct digits that remains prime (leading zeros not allowed) when all occurrences of its digits d are deleted.

Original entry on oeis.org

23, 137, 6173, 37019, 5600239, 476710937, 8192454631, 1645957688093, 78456580281239

Offset: 2

Patrick De Geest, Oct 15 2000

			5600239 is a solution for at least 6 digits because 56239, 560039, 560029, 600239, 500239 and 560023 are all primes.

C. Rivera, PrimePuzzle 110
C. K. Caldwell, Prime Curios! 5600239
C. K. Caldwell, Prime Curios! 476710937.

Cf. A057876-A057882, A051362, A034302-A034305.

More terms from Giovanni Resta, Feb 15 2006

A034304 Nonprime; becomes prime if any digit is deleted (zeros not allowed in the number).

Original entry on oeis.org

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 111, 117, 119, 171, 371, 411, 413, 417, 437, 471, 473, 611, 671, 711, 713, 731, 1379, 1397, 1673, 1739, 1937, 1991, 2233, 2277, 2571, 2577, 2811, 3113, 3131, 3173, 3311, 3317, 3479, 4199, 4331, 4433, 4439

Offset: 1

David W. Wilson

From David A. Corneth, Sep 14 2019: (Start)
Terms can't contain digits of the form 0 (mod 3), 1 (mod 3) and 2 (mod 3) as then one can remove a digit to get a multiple of 3. Classifying digits mod 3 could give further restrictions on the frequency of digits per class.
For example, let (d0, d1, d2) be the frequency of digits from each residue class mod 3 respectively. Then a term can't be of the form (0, 2, 3) as removing a digit from the class 2 (mod 3) gives a multiple of 3. (End)

David A. Corneth, Table of n, a(n) for n = 1..502 (first 200 terms from T. D. Noe, terms n = 201..299 from R. Zumkeller, terms <= 10^11).

Cf. A034302-A034305.

Cf. A010051, A259315.

a034304 n = a034304_list !! (n-1)
a034304_list = map read $ filter (f "") $
               map show $ dropWhile (< 10) a259315_list :: [Integer] where
   f _ "" = True
   f us (v:vs) = a010051' (read (us ++ vs)) == 1 && f (us ++ [v]) vs
-- Reinhard Zumkeller, Jun 24 2015

With[{nn=5000},Select[Complement[Range[10,nn],Prime[Range[ PrimePi[ nn]]]], DigitCount[#,10,0]==0&&And@@PrimeQ[FromDigits/@Subsets[ IntegerDigits[#],{IntegerLength[#]-1}]]&]] (* Harvey P. Dale, Apr 06 2012 *)

Definition corrected by T. D. Noe, Apr 02 2008

A034303 Zeroless primes that become nonprime if any digit is deleted.

Original entry on oeis.org

11, 19, 41, 61, 89, 227, 251, 257, 277, 281, 349, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 1117, 1129, 1171, 1187, 1259, 1289, 1423, 1447, 1453, 1471, 1483, 1543, 1553, 1559, 1583, 1621, 1669, 1721, 1741, 1747

Offset: 1

David W. Wilson

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A034302, A034304, A034305.

Cf. A010051, A038618.

import Data.List (inits, tails)
a034303 n = a034303_list !! (n-1)
a034303_list = filter f $ drop 4 a038618_list where
   f x = all (== 0) $ map (a010051 . read) $
             zipWith (++) (inits $ show x) (tail $ tails $ show x)
-- Reinhard Zumkeller, Dec 17 2011

Select[Prime[Range[5,300]],Union[PrimeQ[FromDigits/@Table[Delete[ IntegerDigits[ #], n],{n,IntegerLength[#]}]]] == {False} && !MemberQ[ IntegerDigits[#],0]&] (* Harvey P. Dale, Jan 09 2014 *)

Definition corrected by T. D. Noe, Apr 02 2008

Name edited by Jon E. Schoenfield, Feb 07 2022

A248642 Odious numbers (A000069) remaining odious if any digit is deleted (zeros allowed).

Original entry on oeis.org

11, 14, 21, 22, 28, 41, 42, 44, 47, 74, 81, 82, 84, 87, 88, 131, 161, 164, 191, 193, 194, 211, 256, 261, 262, 321, 322, 328, 352, 355, 381, 382, 388, 419, 421, 422, 491, 494, 502, 522, 552, 555, 569, 611, 614, 641, 642, 644, 647, 704, 744, 749, 764, 769, 793

Offset: 1

Vladimir Shevelev, Oct 10 2014

An analog of the idea Wilson-Dale: A034302, A051362.

			161 is in the sequence, since the numbers 161,61,11,16 are odious.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000069, A248478, A248644, A248659.

odiousQ:=OddQ[First[DigitCount[#,2]]]&;
Select[Range[1000],odiousQ[#]&&Apply[And,Map[odiousQ[FromDigits[#]]&,Subsets[#,{Length[#]-1}]&[IntegerDigits[#]]]]&] (* Peter J. C. Moses, Oct 10 2014 *)

More terms from Peter J. C. Moses, Oct 10 2014

A057877 a(n) = smallest n-digit prime in A057876.

Original entry on oeis.org

23, 113, 1531, 12239, 111317, 1111219, 11119291, 111111197, 1111113173, 11111133017, 111111189919, 1111111411337, 11111111161177, 111111111263311, 1111111111149119, 11111111111179913, 111111111111118771, 1111111111111751371, 11111111111111111131, 111111111111113129773, 1111111111111111337111

Offset: 2

Patrick De Geest, Oct 15 2000

			1531 gives primes 53, 131 and 151 after dropping digits 1, 5 and 3.

Cf. A057876, A057877, A057878, A057879, A057880, A057881, A057882, A057883, A051362, A034302, A034303, A034304, A034305.

filter:= proc(n) local L,d,Lp;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  for d in convert(L,set) do
    Lp:= subs(d=NULL,L);
    if Lp=[] or Lp[-1] = 0 then return false fi;
    if not isprime(add(Lp[i]*10^(i-1),i=1..nops(Lp))) then return false fi;
  od;
  true
end proc:
Res:= NULL:
for t from 1 to 21 do
  for x from (10^(t+1)-1)/9 by 2 do
    if filter(x) then Res:= Res, x; break fi
  od
od:
Res; # Robert Israel, Jul 13 2018

Do[k = (10^n - 1)/9; While[d = IntegerDigits[k]; !PrimeQ[k] || !PrimeQ[ FromDigits[ DeleteCases[d, 0]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 1]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 2]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 3]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 4]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 5]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 6]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 7]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 8]]] || !PrimeQ[ FromDigits[ DeleteCases[d, 9]]], k++ ]; Print[k], {n, 2, 19}]

Extended by Robert G. Wilson v, Dec 17 2002

More terms from Robert Israel, Jul 13 2018

A057880 Primes with 4 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of its digits d.

Original entry on oeis.org

6173, 12239, 16673, 19531, 19973, 21613, 22397, 22937, 34613, 36137, 47933, 51193, 54493, 56519, 56531, 56591, 69491, 69497, 72937, 76873, 93497, 96419, 96479, 96497, 98837, 112939, 118213, 131779, 143419, 144497, 159319, 163337

Offset: 1

Patrick De Geest, Oct 15 2000

Robert Israel, Table of n, a(n) for n = 1..653

Cf. A057876-A057883, A051362, A034302-A034305.

filter:= proc(L) local d,Lp,i;
      if L[-1]=0 then return false fi;
      if not isprime(add(L[i]*10^(i-1),i=1..nops(L))) then return false fi;
      for d in convert(L,set) do
        Lp:= remove(`=`,L,d);
        if Lp[-1] = 0 or not isprime(add(Lp[i]*10^(i-1),i=1..nops(Lp))) then return false fi;
      od;
      true
end proc:
getCands:= proc(n, m) option remember;
   if m = 1 then return [seq([d$n], d=0..9)] fi;
   if n < m then return [] fi;
   [seq(seq([i,op(L)],i= {$0..9} minus convert(L,set)),L = procname(n-1,m-1)),
    seq(seq([i,op(L)],i=convert(L,set)),L = procname(n-1,m))]
end proc:
[seq(op(sort(map(t->add(t[i]*10^(i-1),i=1..nops(t)),select(filter,getCands(d,4))))),d=4..6)]; # Robert Israel, Jan 19 2017

p4dQ[n_]:=Module[{idn=IntegerDigits[n]},Count[idn,0]==0 && Count[ DigitCount[ n],0]==6&&AllTrue[FromDigits/@Table[DeleteCases[idn,k],{k,Union[idn]}],PrimeQ]]; Select[Prime[Range[ 15000]],p4dQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 30 2017 *)

Offset changed by Robert Israel, Jan 19 2017

A057882 Primes with 6 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of its digits d.

Original entry on oeis.org

5600239, 21066319, 42209639, 63019679, 82190131, 95422517, 113420491, 114248737, 130194791, 132863191, 135160339, 137697019, 145136591, 145611439, 146414839, 153160517, 159136291, 181680713, 186601339, 186609331, 190714133

Offset: 1

Patrick De Geest, Oct 15 2000

Cf. A057876-A057883, A051362, A034302-A034305.

Offset changed by Andrew Howroyd, Aug 14 2024

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A051362 Primes remaining prime if any digit is deleted (zeros allowed).

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A034305 Zeroless nonprimes that remain nonprime if any digit is deleted.

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A057876 Primes p with the following property: let d_1, d_2, ... be the distinct digits occurring in the decimal expansion of p. Then for each d_i, dropping all the digits d_i from p produces a prime number. Leading 0's are not allowed.

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A057883 Smallest possible prime with at least n (from 2 to 10) distinct digits that remains prime (leading zeros not allowed) when all occurrences of its digits d are deleted.

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A034304 Nonprime; becomes prime if any digit is deleted (zeros not allowed in the number).

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A034303 Zeroless primes that become nonprime if any digit is deleted.

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A248642 Odious numbers (A000069) remaining odious if any digit is deleted (zeros allowed).

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A057877 a(n) = smallest n-digit prime in A057876.

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