A158125 -id:A158125 - cp's OEIS frontend

A050249 Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, 5152507, 5564453, 5575259, 6173731, 6191371, 6236179, 6463267, 6712591, 7204777, 7469789, 7469797

Offset: 1

Eric W. Weisstein

Tao proved that this sequence is infinite. - T. D. Noe, Mar 01 2011

For k = 5, 6, 7, 8, 9, 10, the number of terms < 10^k in this sequence is 0, 5, 35, 334, 3167, 32323. - Jean-Marc Rebert, Nov 10 2015

Michael Filaseta and Jeremiah Southwick, Primes that become composite after changing an arbitrary digit, Math. Comp. (2021) Vol. 90, 979-993. doi:10.1090/mcom/3593

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..1317 from Klaus Brockhaus, terms 1318..3167 from Jean-Marc Rebert).
Michael Filaseta and Jacob Juillerat, Consecutive primes which are widely digitally delicate, arXiv:2101.08898 [math.NT], 2021.
Jon Grantham, Finding a Widely Digitally Delicate Prime, arXiv:2109.03923 [math.NT], 2021.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Jackson Hopper and Paul Pollack, Digitally delicate primes, arXiv:1510.03401 [math.NT], 2015.
Dana Jacobsen, Digitally delicate primes up to 1e11
Matt Parker, How do you prove a prime is infinitely fragile?, Stand-up Maths YouTube video, 2022.
Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).
Terence Tao, A remark on primality testing and decimal expansions, arXiv:0802.3361 [math.NT], 2008-2010; Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
Eric Weisstein's World of Mathematics, Weakly Prime

Cf. A118118, A158124 (weakly primes), A158125 (weakly primes).

Cf. A137985 (analogous base-2 sequence), A186995 (weak primes in base n).

IsA118118:=function(n); D:=Intseq(n); return forall{ : k in [1..#D], j in [0..9] | j eq D[k] or not IsPrime(Seqint(S)) where S:=Insert(D, k, k, [j]) }; end function; [ p: p in PrimesUpTo(8000000) | IsA118118(p) ]; // Klaus Brockhaus, Feb 28 2011

fQ[n_] := Block[{d = IntegerDigits@ n, t = {}}, Do[AppendTo[t, FromDigits@ ReplacePart[d, i -> #] & /@ DeleteCases[Range[0, 9], x_ /; x == d[[i]]]], {i, Length@ d}]; ! AnyTrue[Flatten@ t, PrimeQ]] ; Select[Prime@ Range[10^5], fQ] (* Michael De Vlieger, Nov 10 2015, Version 10 *)

isokp(n) = {v = digits(n); for (k=1, #v, w = v; for (j=0, 9, if (j != v[k], w[k] = j; ntest = subst(Pol(w), x, 10); if (isprime(ntest), return(0));););); return (1);}
lista(nn) = {forprime(p=2, nn, if (isokp(p), print1(p, ", ")););} \\ Michel Marcus, Dec 15 2015

from sympy import isprime
def h1(n): # hamming distance 1 neighbors of n
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i])
def ok(n): return isprime(n) and all(not isprime(k) for k in h1(n) if k!=n)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jun 19 2022

Edited by Charles R Greathouse IV, Aug 02 2010

A158124 Weakly prime numbers (or isolated primes): changing any one decimal digit always produces a composite number, with restriction that first digit may not be changed to a 0.

Original entry on oeis.org

294001, 505447, 584141, 604171, 929573, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3070663, 3085553, 3326489, 4393139, 5152507, 5285767, 5564453, 5575259, 5974249, 6173731, 6191371, 6236179, 6463267, 6712591, 7204777, 7469789, 7469797, 7810223

Offset: 1

Eric W. Weisstein, Mar 13 2009

The definition could be restated as "primes p with d digits such that there is no prime q with at most d digits at Hamming distance 1 from p (in base 10)". - N. J. A. Sloane, May 06 2019

For the following values of k, 5, 6, 7, 8, 9, 10, the number of terms < 10^k in this sequence is 0, 6, 43, 406, 3756, 37300. - Jean-Marc Rebert, Nov 10 2015

Jean-Marc Rebert, Table of n, a(n) for n = 1..3756
C. Rivera, Weakly Primes
Eric Weisstein's World of Mathematics, Weakly Prime

Cf. A050249, A158125 (weakly primes), A186995, A192545.

filter:= proc(n)
  local L,i,d,ds;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  for i from 1 to nops(L) do
    if i = nops(L) then ds:= {$1..9} minus {L[i]}
    elif i = 1 then ds:= {1,3,7,9} minus {L[i]}
    else ds:= {$0..9} minus {L[i]}
    fi;
    for d in ds do
      if isprime(n + (d - L[i])*10^(i-1)) then return false fi;
    od
  od;
  true
end proc:
select(filter, [seq(i,i=11..10^6,2)]); # Robert Israel, Dec 15 2015

Select[Prime@ Range[10^5], Function[n, Function[w, Total@ Map[Boole@ PrimeQ@ # &, DeleteCases[#, n]] &@ Union@ Flatten@ Map[Function[d, FromDigits@ ReplacePart[w, d -> #] & /@ If[d == 1, #, Prepend[#, 0]] &@ Range@ 9], Range@ Length@ w] == 0]@ IntegerDigits@ n]] (* Michael De Vlieger, Dec 13 2016 *)

isokp(n) = {v = digits(n); for (k=1, #v, w = v; if (k==1, idep = 1, idep=0); for (j=idep, 9, if (j != v[k], w[k] = j; ntest = subst(Pol(w), x, 10); if (isprime(ntest), return(0));););); return (1);}
lista(nn) = {forprime(p=2, nn, if (isokp(p), print1(p, ", ")););} \\ Michel Marcus, Dec 15 2015

from sympy import isprime
def h1(n): # hamming distance 1 neighbors of n, not starting with 0
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i] and not (i==0 and c=="0"))
def ok(n): return isprime(n) and all(not isprime(k) for k in h1(n))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 31 2022

Edited by Charles R Greathouse IV, Aug 02 2010

Missing a(3385) inserted into b-file by Andrew Howroyd, Feb 23 2018

A192545 Numbers such that all numbers are composite when replacing exactly one digit with another, except the leading digit with zero.

Original entry on oeis.org

200, 202, 204, 205, 206, 208, 320, 322, 324, 325, 326, 328, 510, 512, 514, 515, 516, 518, 530, 532, 534, 535, 536, 538, 620, 622, 624, 625, 626, 628, 840, 842, 844, 845, 846, 848, 890, 892, 894, 895, 896, 898, 1070, 1072, 1074, 1075, 1076, 1078, 1130, 1132

Offset: 1

Reinhard Zumkeller, Jul 05 2011

A048853(a(n)) = 0;
Intersection of this sequence and A000040 is A158124. - Evgeny Kapun, Dec 13 2016
If the last digit of an element is 0, 2, 4, 5, 6 or 8, then replacing it with 0, 2, 4, 5, 6 or 8 also yields an element. - David A. Corneth and corrected by Evgeny Kapun, Dec 13 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequences: A050249, A118118, A143641, A158124, A158125.

import Data.List (elemIndices)
a192545 n = a192545_list !! (n-1)
a192545_list = map (+ 1) $ elemIndices 0 $ map a048853 [1..]

Select[Range@ 1200, Function[w, Total@ Boole@ Flatten@ Map[Function[d, PrimeQ@ FromDigits@ ReplacePart[w, d -> #] & /@ If[d == 1, #, Prepend[#, 0]] &@ Range@ 9], Range@ Length@ w] == 0]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 13 2016 *)

A158641 Strong primes p: adding 2 to any one digit of p produces a prime number (no digits 8 & 9 in p).

Original entry on oeis.org

3, 5, 11, 17, 41, 107, 137, 347, 2111, 2657, 3527, 4421, 6761, 21011, 24371, 32057

Offset: 1

Zak Seidov, Mar 23 2009

All terms are lesser of twin pairs.

The sequence is finite. In particular, there are no terms longer than 5 digits since 2*10^k for k=0,1,...,5 give all nonzero residues modulo 7, implying that adding 2 to some digit yields a multiple of 7. - Max Alekseyev, Apr 25 2010

			2111 is in this sequence because all 2111, 4111, 2311, 2131 and 2113 are prime numbers.
32057 is in this sequence because all 32057, 52057, 34057, 32257, 32077 and 32059 are prime numbers.

Cf. A050249, A158124, A158125 Weakly prime numbers (changing any one digit always produces a composite number).

Lton := proc(L) local i; add(op(i,L)*10^(i-1),i=1..nops(L)) ; end: isA158641 := proc(p) local pdgs,pplus,i ; if isprime(p) then pdgs := convert(p,base,10) ; if convert(pdgs,set) intersect {8,9} <> {} then false; else for i from 1 to nops(pdgs) do pplus := subsop(i=2+op(i,pdgs),pdgs) ; if not isprime(Lton(pplus)) then RETURN(false); fi; od: true; fi; else false; fi; end: for n from 1 do p := ithprime(n) ; if isA158641(p) then print(p) ; fi; od: # R. J. Mathar, Apr 16 2009

spQ[p_]:=Max[IntegerDigits[p]]<8&&AllTrue[FromDigits/@Table[MapAt[ 2+#&,IntegerDigits[ p],n],{n,IntegerLength[p]}],PrimeQ]; Select[Prime[ Range[ 3500]],spQ] (* Harvey P. Dale, Nov 26 2022 *)

test(p)=my(v=eval(Vec(Str(p)))); for(i=1,#v, if(v[i]>7,return(0))); for(i=0,#v-1, if(!isprime(p+2*10^i), return(0))); 1
forprime(p=2,4e9, if(isprime(p+2) && test(p), print1(p","))) \\ Charles R Greathouse IV, Sep 09 2009

has(n)=if(vecmax(Set(digits(n)))>7, return(0)); for(i=0,#digits(n)-1, if(!isprime(n+2*10^i), return(0))); 1
select(has, primes(3438)) \\ Charles R Greathouse IV, Mar 11 2016

Keywords full, fini from Max Alekseyev, Apr 25 2010

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A050249 Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.

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A158124 Weakly prime numbers (or isolated primes): changing any one decimal digit always produces a composite number, with restriction that first digit may not be changed to a 0.

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A192545 Numbers such that all numbers are composite when replacing exactly one digit with another, except the leading digit with zero.

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A158641 Strong primes p: adding 2 to any one digit of p produces a prime number (no digits 8 & 9 in p).

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Maple

Mathematica

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PARI

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