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A050249
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Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.
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16
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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
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OFFSET
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1,1
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COMMENTS
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Tao proved that this sequence is infinite. - T. D. Noe, Mar 01 2011
For the following values 5, 6, 7, 8, 9, 10 of k, the number of terms < 10^k in this sequence is 0, 5, 35, 334, 3167, 32323. - Jean-Marc Rebert, Nov 10 2015
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LINKS
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Klaus Brockhaus and Jean-Marc Rebert, Table of n, a(n) for n = 1..3167 (first 1317 terms from Klaus Brockhaus)
Jackson Hopper, Paul Pollack, Digitally delicate primes, arXiv:1510.03401 [math.NT], 2015.
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
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MATHEMATICA
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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 *)
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PROG
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(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
(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); }
lista(nn) = {forprime(p=2, nn, if (isokp(p), print1(p, ", ")); ); } \\ Michel Marcus, Dec 15 2015
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CROSSREFS
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Cf. A118118, A158124 (weakly primes), A158125 (weakly primes)
Cf. A137985 (analogous base 2 sequence), A186995 (weak primes in base n).
Sequence in context: A254843 A318787 A158124 * A224973 A328664 A328935
Adjacent sequences: A050246 A050247 A050248 * A050250 A050251 A050252
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KEYWORD
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nonn,base
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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Edited by Charles R Greathouse IV, Aug 02 2010
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STATUS
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approved
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