A034305 -id:A034305 - cp's OEIS frontend

A034302 Zeroless primes that remain prime if any digit is deleted.

23, 37, 53, 73, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1499, 1997, 2239, 2293, 3137, 4919, 6173, 7433, 9677, 19973, 23833, 26833, 47933, 73331, 74177, 91733, 93491, 94397, 111731, 166931, 333911, 355933, 477797, 477977

Offset: 1

David W. Wilson

Chai Wah Wu, Table of n, a(n) for n = 1..118 (terms 1..79 from T. D. Noe, terms 80..103 from Charles R Greathouse IV)
StackExchange, Deleting any digit yields a prime

Cf. A034303, A034304, A034305, A051362, A010051, A038618.

import Data.List (inits, tails)
a034302 n = a034302_list !! (n-1)
a034302_list = filter f $ drop 4 a038618_list where
   f x = all (== 1) $ map (a010051 . read) $
             zipWith (++) (inits $ show x) (tail $ tails $ show x)
-- Reinhard Zumkeller, Dec 17 2011

rpnzQ[n_]:=Module[{idn=IntegerDigits[n]},Count[idn,0]==0 && And@@ PrimeQ[FromDigits/@ Subsets[IntegerDigits[n], {Length[idn]-1}]]]; Select[Prime[Range[40000]],rpnzQ]  (* Harvey P. Dale, Mar 24 2011 *)

is(n)=my(d=digits(n),t=2^#d-1); if(vecmin(d)==0, return(0)); for(i=0,#d-1, if(!isprime(fromdigits(vecextract(d,t-2^i))), return(0))); isprime(n) \\ Charles R Greathouse IV, Jun 23 2017

from itertools import product
from sympy import isprime
A034302_list, m = [23, 37, 53, 73], 7
for l in range(1,m-1): # generate all terms less than 10^m
    for d in product('123456789',repeat=l):
        for e in product('1379',repeat=2):
            s = ''.join(d+e)
            if isprime(int(s)):
                for i in range(len(s)):
                    if not isprime(int(s[:i]+s[i+1:])):
                        break
                else:
                    A034302_list.append(int(s)) # Chai Wah Wu, Apr 05 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

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

A057878 Primes with 2 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of its digits d.

Original entry on oeis.org

23, 37, 53, 73, 113, 131, 151, 211, 311, 11111111111111111131, 11111111111111117111, 11111111111131111111, 11111111131111111111, 111111111111111111111113, 111111111111111112111111, 111111111111111121111111, 111111111112111111111111, 111111115111111111111111

Offset: 1

Patrick De Geest, Oct 15 2000

Digits 0, 4, 6, 8, and 9 can never occur; digits 2, 3, 5, 7 can occur at most once in a term; every other digit is a 1. - Sean A. Irvine, Jul 11 2022

Cf. A057876-A057883, A004022, A004023, A051362, A034302-A034305.

More terms from Sean A. Irvine, Jul 11 2022

A057879 Primes with 3 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of any one of its distinct digits.

Original entry on oeis.org

137, 173, 179, 197, 317, 431, 617, 719, 1531, 1831, 1997, 2113, 2131, 2237, 2273, 2297, 2311, 2797, 3137, 3371, 4337, 4373, 4733, 4919, 7297, 7331, 7573, 7873, 8191, 8311, 8831, 8837, 33413, 33713, 34313, 37313, 41117, 41999, 44417, 49199, 73331

Offset: 1

Patrick De Geest, Oct 15 2000

Numbers in A057876 with exactly 3 distinct digits.

Intersection of A057876 and A235155.

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

Offset changed by Andrew Howroyd, Aug 14 2024

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A034302 Zeroless primes that remain prime 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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A057877 a(n) = smallest n-digit prime in A057876.

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

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

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A057878 Primes with 2 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of its digits d.

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