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 A068164 Smallest prime obtained from n by inserting zero or more decimal digits. 10
 11, 2, 3, 41, 5, 61, 7, 83, 19, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 1201, 211, 223, 23, 241, 251, 263, 127, 281, 29, 307, 31, 1321, 233, 347, 353, 367, 37, 383, 139, 401, 41, 421, 43, 443, 457, 461, 47, 487, 149, 503, 151, 521, 53, 541, 557, 563, 157 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The digits may be added before, in the middle of, or after the digits of n. a(n) = n for prime n, by definition. - Zak Seidov, Nov 13 2014 a(n) always exists. Proof. Suppose n is L digits long, and let n' = 10*n+1. The arithmetic progression k*10^(2L)+n' (k >= 0) contains infinitely many primes, by Dirichlet's theorem, and they all contain the digits of n. QED. - Robert Israel, Nov 13 2014. For another proof, see A018800. Similar to but different from A062584. E.g. a(133) = 1033, but A062584(133) = 4133. LINKS Allan C. Wechsler, Table of n, a(n) for n = 1..1000 EXAMPLE Smallest prime formed from 20 is 1201, by placing 1 on both sides. Smallest prime formed from 33 is 233, by placing a 2 in front. MAPLE A068164 := proc(n)     local p, pdigs, plen, dmas, dmasdigs, i, j;     # test all primes ascending     p := 2;     while true do         pdigs := convert(p, base, 10) ;         plen := nops(pdigs) ;         # binary digit mask over p         for dmas from 2^plen-1 to 0 by -1 do             dmasdigs := convert(dmas, base, 2) ;             pdel := [] ;             for i from 1 to nops(dmasdigs) do                 if op(i, dmasdigs) = 1 then                     pdel := [op(pdel), op(i, pdigs)] ;                 end if;             end do:             if n = add(op(j, pdel)*10^(j-1), j=1..nops(pdel)) then                 return p;             end if;         end do:         p := nextprime(p) ;     end do: end proc: seq(A068164(n), n=1..120) ; # R. J. Mathar, Nov 13 2014 PROG (Haskell) a068164 n = head (filter isPrime (digitExtensions n)) digitExtensions n = filter (includes n) [0..] includes n k = listIncludes (show n) (show k) listIncludes [] _ = True listIncludes (h:_) [] = False listIncludes l1@(h1:t1) (h2:t2) = if (h1 == h2) then (listIncludes t1 t2) else (listIncludes l1 t2) isPrime 1 = False isPrime n = not (hasDivisorAtLeast 2 n) hasDivisorAtLeast k n = (k*k <= n) && (((n `rem` k) == 0) || (hasDivisorAtLeast (k+1) n)) CROSSREFS Cf. A018800 (an upper bound), A060386, A062584 (also an upper bound). Cf. also A068165. Sequence in context: A089744 A160137 A107698 * A089754 A110743 A077549 Adjacent sequences:  A068161 A068162 A068163 * A068165 A068166 A068167 KEYWORD base,easy,nonn AUTHOR Amarnath Murthy, Feb 25 2002 EXTENSIONS Corrected by Ray Chandler, Oct 11 2003 Haskell code and b-file added by Allan C. Wechsler, Nov 13 2014 STATUS approved

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Last modified September 28 01:20 EDT 2020. Contains 337388 sequences. (Running on oeis4.)