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A050249 Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Eric W. Weisstein

EXTENSIONS

Edited by Charles R Greathouse IV, Aug 02 2010

STATUS

approved

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Last modified January 17 14:16 EST 2021. Contains 340245 sequences. (Running on oeis4.)