A051362 -id:A051362 - cp's OEIS frontend

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

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

A057881 Primes with 5 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of its digits d.

Original entry on oeis.org

37019, 159013, 198013, 210139, 223697, 226397, 236297, 1305593, 1388693, 1393697, 1900937, 1912831, 2370673, 2796337, 2882093, 2930773, 3200191, 3202139, 3346199, 3442693, 3463199, 3463619, 3746399, 3769133, 4234039

Offset: 1

Patrick De Geest, Oct 15 2000

Numbers in A057876 with exactly 5 distinct digits.

Intersection of A057876 and A235157.

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

Offset changed by Andrew Howroyd, Aug 14 2024

A178423 Semiprimes for which dropping any digit gives a prime number.

Original entry on oeis.org

22, 25, 33, 35, 55, 57, 77, 111, 119, 371, 411, 413, 417, 437, 471, 473, 611, 671, 713, 731, 1037, 1073, 1079, 1379, 1397, 1673, 1739, 1937, 1991, 2571, 2577, 2811, 3113, 3131, 3173, 3317, 4331, 4439, 4499, 4631, 6017, 6431, 6773, 7619, 9977, 12777

Offset: 1

Jonathan Vos Post, May 27 2010

This is the 2nd row of the infinite array A[k,n] = n-th number with k prime factors (not necessarily distinct) for which dropping any digit gives a prime number.
The first row A[1,n] = A051362 = numbers n such that n remains prime if any digit is deleted (zeros allowed).
The 3rd row A[3,n] begins {27 = 3^3, 52 = 2^2 * 13, 75 = 3 * 5^2, 117 = 3^2 * 13, 171 = 3^2 * 19, ...}.
The 4th row A[4,n] begins: {2277 = 3^2 * 11 * 23, 5577 = 3 * 11 * 13^2, 8211 = 3 * 7 * 17 * 23, 8811 = 3^2 * 11 * 89, ...}.
The 5th row A[5,n] begins:{32 = 2^5, 72 = 2^3 x 3^2, ...}.

			a(9) = 119 because this is a semiprime (119 = 7 * 17), dropping the leftmost digit gives 19 (a prime), dropping the middle digit gives 19 (a prime), and dropping the rightmost digit gives 11 (a prime).

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A000040, A001358, A034895, A108632.

ddp[n_]:=Module[{idn=IntegerDigits[n]},PrimeOmega[n]==2 && And@@PrimeQ[ FromDigits/@Table[Drop[idn,{i}],{i,Length[idn]}]]]; Select[Range[ 13000],ddp] (* Harvey P. Dale, Apr 10 2012 *)

A001358 INTERSECTION A034895.

A248745 Primes becoming a composite number if any digit is deleted (zeros allowed).

Original entry on oeis.org

89, 227, 251, 257, 277, 281, 349, 409, 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, 1777, 1847, 1889

Offset: 1

Vladimir Shevelev, Oct 13 2014

Since J. Gutierrez called terms of A051362 "super-prime numbers", then it is natural to call the terms of this sequence "weakest primes".

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

Cf. A051362, A000040.

compositeQ[n_]:=!(Abs[n]==1||PrimeQ[n]);
Select[Prime[Range[5,500]],Apply[And,Map[compositeQ[FromDigits[#]]&,Subsets[#,{Length[#]-1}]&[IntegerDigits[#]]]]&] (* Peter J. C. Moses, Oct 13 2014 *)
Select[Prime[Range[300]],AllTrue[FromDigits/@Table[Drop[IntegerDigits[#],{k}],{k,IntegerLength[#]}],CompositeQ]&] (* Harvey P. Dale, Oct 30 2021 *)

More terms from Peter J. C. Moses, Oct 13 2014

A267413 Dropping any binary digit gives a prime number.

Original entry on oeis.org

6, 7, 11, 15, 35, 39, 63, 135, 255, 999, 2175, 8223, 16383, 57735, 131075, 131079, 262143, 524295, 1048575, 536870919, 1073735679, 2147483655, 4294967295, 17179770879, 4260641103903, 4611686018427387903, 4720069647059686260735, 1237940039285380274899124223

Offset: 1

Stanislav Sykora, Jan 14 2016

This is the binary analog of A034895. The sequence contains mostly numbers with very few binary digit runs (BDR, A005811). Those with one BDR are of the type 2^k-1, such that 2^(k-1)-1 is a Mersenne prime (A000668). Vice versa, if M is any Mersenne prime, then 2*M+1 is a term. The number 6 is the only term with an even number of BDRs. There are many terms with 3 BDRs. The first term with 5 BDRs is 57735. The next terms with at least 5 BDRs (if they exist at all) are larger than 10^10. So far, I could test that a(24) > 10^10.
From Robert Israel, Jan 14 2016: (Start)
For n >= 2, a(n) == 3 (mod 4).
2^k+3 is in the sequence if 2^(k-1)+1 and 2^(k-1)+3 are primes, i.e., 2^(k-1)+1 is in the intersection of A019434 and A001359.  The only known terms of the sequence in this class are 7, 11, 35, 131075.
2^k+7 is in the sequence if 2^(k-1)+3 and 2^(k-1)+7 are primes: thus 2^(k-1)+3 is in A057733 and 2^(k-1)+7 is in A104066.  Terms of the sequence in this class include 15, 39, 135, 131079, 524295, 536870919, 2147483655 (but no more for k <= 2000).
(End)
a(25) > 2^38. - Giovanni Resta, Apr 10 2016
For n > 1, a(n) = 2p+1 for some prime p. - Chai Wah Wu, Aug 27 2021
From Bert Dobbelaere, Aug 07 2023: (Start)
There are no more terms with an odd number of binary digits: from any number having an odd number of binary digits, one can always drop a digit and obtain a multiple of 3. Numbers of the form 2^k+3 (k even and k > 2) cannot be terms because 2^(k-1)+1 is a multiple of 3.
(End)

			Decimal and binary forms of the known terms:
   1           6                                110
   2           7                                111
   3          11                               1011
   4          15                               1111
   5          35                             100011
   6          39                             100111
   7          63                             111111
   8         135                           10000111
   9         255                           11111111
  10         999                         1111100111
  11        2175                       100001111111
  12        8223                     10000000011111
  13       16383                     11111111111111
  14       57735                   1110000110000111 <--- (a binary palindrome
  15      131075                 100000000000000011       with 5 digit runs)
  16      131079                 100000000000000111
  17      262143                 111111111111111111
  18      524295               10000000000000000111
  19     1048575               11111111111111111111
  20   536870919     100000000000000000000000000111
  21  1073735679     111111111111111110011111111111
  22  2147483655   10000000000000000000000000000111
  23  4294967295   11111111111111111111111111111111
  24 17179770879 1111111111111111100111111111111111

Cf. A000668, A001359, A005811, A019434, A034895 (base 10), A051362, A057733, A104066.

filter:= proc(n) local B,k,y;
   if not isprime(floor(n/2)) then return false fi;
   B:= convert(n,base,2);
   for k from 2 to nops(B) do
     if B[k] <> B[k-1] then
       y:= n mod 2^(k-1);
       if not isprime((y+n-B[k]*2^(k-1))/2) then return false fi
     fi
   od;
   true
end proc:
select(filter, [6, seq(i,i=7..10^6,4)]); # Robert Israel, Jan 14 2016

Select[Range[2^20], AllTrue[Function[w, Map[FromDigits[#, 2] &@ Drop[w, {#}] &, Range@ Length@ w]]@ IntegerDigits[#, 2], PrimeQ] &] (* Michael De Vlieger, Jan 16 2016, Version 10 *)

DroppingAnyDigitGivesAPrime(N,b) = {
\\ Property-testing function; returns 1 if true for N, 0 otherwise
\\ Works with any base b. Here used with b=2.
  my(k=b,m); if(N=(k\b), m=(N\k)*(k\b)+(N%(k\b));
    if ((m<2)||(!isprime(m)),return(0)); k*=b);
  return(1);
}

from sympy import isprime
def ok(n):
    if n < 7 or n%4 != 3: return n == 6
    b = bin(n)[2:]
    return all(isprime(int(b[:i]+b[i+1:], 2)) for i in range(len(b)))
print(list(filter(ok, range(2, 2**20)))) # Michael S. Branicky, Jun 07 2021

a(24) from Giovanni Resta, Apr 10 2016

a(25)-a(28) from Bert Dobbelaere, Aug 07 2023

A378415 Primes with repeated digits that remain prime when any two of the same-valued digits are deleted.

Original entry on oeis.org

113, 131, 151, 211, 223, 227, 233, 277, 311, 337, 353, 373, 443, 557, 577, 599, 727, 733, 757, 773, 883, 887, 929, 997, 1009, 1013, 1021, 1031, 1051, 1103, 1117, 1123, 1129, 1153, 1171, 1213, 1223, 1229, 1231, 1291, 1373, 1399, 1447, 1471, 1531, 1553, 1559, 1663, 1667, 1669, 1733, 1777

Offset: 1

Enrique Navarrete, Nov 25 2024

Relaxed version of A378081, which contains only 18 terms up to 10^100.

Not a superset of A378081 since this sequence does not contain 257 and 523.

			114217 is in the sequence since deleting any two of the three 1's gives 4217 and 1427, both of which are prime.
131371 is not in the sequence since deleting the two 3's gives 1171, which is prime, but deleting two of the three 1's gives 3371, 3137, and 1337, the last one of which is not prime.

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

Cf. A051362, A226108, A378081.

from sympy import isprime
from itertools import combinations as C
def ok(n):
    if n<100  or not isprime(n) or len(s:=str(n))==len(set(s)): return False
    return all(isprime(int(t)) for i, j in C(range(len(s)), 2) if s[i]==s[j] and (t:=s[:i]+s[i+1:j]+s[j+1:])!="")
print([k for k in range(1800) if ok(k)]) # Michael S. Branicky, Nov 25 2024

A378563 Primes that remain prime if any three of their digits are deleted.

Original entry on oeis.org

2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557, 5573, 5737, 7237, 7253, 7333, 7523, 7537, 7573, 7577, 7723, 7727, 7753, 7757, 11113, 11117, 11119, 11131, 11171, 11173, 11197

Offset: 1

Enrique Navarrete, Nov 30 2024

Any 4-digit term has all digits prime (cf. A019546).
The corresponding sequence for two digits deleted contains only 18 terms up to 10^100 (Cf. A378081).
Any term >= 10000 must have its last four digits be from {1, 3, 7, 9}. - Michael S. Branicky, Dec 01 2024

			43117 is in the sequence since upon deleting any three digits we get 43, 31, 11, 17 and 47, all of which are prime.

Cf. A019546, A051362, A378081.

from sympy import isprime
from itertools import combinations as C
def ok(n):
    if n < 1000 or not isprime(n): return False
    s = str(n)
    return all(isprime(int(t)) for i, j, k in C(range(len(s)), 3) if (t:=s[:i]+s[i+1:j]+s[j+1:k]+s[k+1:])!="")
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Dec 01 2024

cp's OEIS Frontend

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

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A057881 Primes with 5 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of its digits d.

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A178423 Semiprimes for which dropping any digit gives a prime number.

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A248745 Primes becoming a composite number if any digit is deleted (zeros allowed).

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A267413 Dropping any binary digit gives a prime number.

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A378415 Primes with repeated digits that remain prime when any two of the same-valued digits are deleted.

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