cp's OEIS Frontend

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

%I A050249 #103 Feb 16 2025 08:32:40
%S A050249 294001,505447,584141,604171,971767,1062599,1282529,1524181,2017963,
%T A050249 2474431,2690201,3085553,3326489,4393139,5152507,5564453,5575259,
%U A050249 6173731,6191371,6236179,6463267,6712591,7204777,7469789,7469797
%N A050249 Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.
%C A050249 Tao proved that this sequence is infinite. - _T. D. Noe_, Mar 01 2011
%C A050249 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
%D A050249 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
%H A050249 Jon E. Schoenfield, <a href="/A050249/b050249.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1317 from Klaus Brockhaus, terms 1318..3167 from Jean-Marc Rebert).
%H A050249 Michael Filaseta and Jacob Juillerat, <a href="https://arxiv.org/abs/2101.08898">Consecutive primes which are widely digitally delicate</a>, arXiv:2101.08898 [math.NT], 2021.
%H A050249 Jon Grantham, <a href="https://arxiv.org/abs/2109.03923">Finding a Widely Digitally Delicate Prime</a>, arXiv:2109.03923 [math.NT], 2021.
%H A050249 Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H A050249 Jackson Hopper and Paul Pollack, <a href="http://arxiv.org/abs/1510.03401">Digitally delicate primes</a>, arXiv:1510.03401 [math.NT], 2015.
%H A050249 Dana Jacobsen, <a href="https://ntheory.org/data/delicate_1e11.txt">Digitally delicate primes up to 1e11</a>
%H A050249 Matt Parker, <a href="https://www.youtube.com/watch?v=p3Khnx0lUDE">How do you prove a prime is infinitely fragile?</a>, Stand-up Maths YouTube video, 2022.
%H A050249 Jeremiah T. Southwick, <a href="https://scholarcommons.sc.edu/etd/5879/">Two Inquiries Related to the Digits of Prime Numbers</a>, Ph. D. Dissertation, University of South Carolina (2020).
%H A050249 Terence Tao, <a href="http://arxiv.org/abs/0802.3361">A remark on primality testing and decimal expansions</a>, arXiv:0802.3361 [math.NT], 2008-2010; Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
%H A050249 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WeaklyPrime.html">Weakly Prime</a>
%t A050249 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 *)
%o A050249 (Magma) IsA118118:=function(n); D:=Intseq(n); return forall{ <k, j>: 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
%o A050249 (PARI) 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);}
%o A050249 lista(nn) = {forprime(p=2, nn, if (isokp(p), print1(p, ", ")););} \\ _Michel Marcus_, Dec 15 2015
%o A050249 (Python)
%o A050249 from sympy import isprime
%o A050249 def h1(n): # hamming distance 1 neighbors of n
%o A050249     s = str(n); d = "0123456789"; L = len(s)
%o A050249     yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i])
%o A050249 def ok(n): return isprime(n) and all(not isprime(k) for k in h1(n) if k!=n)
%o A050249 print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Jun 19 2022
%Y A050249 Cf. A118118, A158124 (weakly primes), A158125 (weakly primes).
%Y A050249 Cf. A137985 (analogous base-2 sequence), A186995 (weak primes in base n).
%K A050249 nonn,base
%O A050249 1,1
%A A050249 _Eric W. Weisstein_
%E A050249 Edited by _Charles R Greathouse IV_, Aug 02 2010