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A258706 Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown. 9

%I

%S 2,3,5,7,11,13,17,37,79,113,199,337,1111111111111111111,

%T 11111111111111111111111

%N Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown.

%C See the main entry, A003459, for further information and references cited below.

%C The next terms are the repunit primes (A004023) R(317), too large to be displayed here, and R(1031), too large even for a b-file. Johnson (1977) proves that subsequent terms must be of the form a*R(n) + b*10^k, with a and a+b in {1..9}, k < n, and n > 9*10^9 if b != 0. - _M. F. Hasler_, Jun 26 2018

%H M. F. Hasler, <a href="/A258706/b258706.txt">Table of n, a(n) for n = 1..15</a>

%t Flatten@{2, 3, 5, 7,

%t Table[Select[

%t Table @@

%t Prepend[Prepend[

%t Table[{A@k, A[k - 1], 4}, {k, 2, n}], {A[1], 4}],

%t Unevaluated[

%t Unevaluated[FromDigits[{1, 3, 7, 9}[[A /@ Range[n]]]]]]] //

%t Flatten,

%t Function[L,

%t And[PrimeQ[#],

%t And @@ PrimeQ[

%t FromDigits /@ (Permute[L, #] & /@

%t RandomPermutation[Length@L, 5])],

%t And @@ PrimeQ[FromDigits /@ Rest[Permutations[L]]]]]@

%t IntegerDigits@# &], {n, 2, 33}]}

%t (* Exhaustively searches thru 33 digits in ~7.5 sec, and up to 69 digits in 5 min, but cannot reach 317 digits. Not helpful in the light of Schroeppel's theorem that it's all repunits past 991. - _Bill Gosper_, Jan 06 2017 *)

%o (Haskell)

%o import Data.List (permutations, (\\))

%o a258706 n = a258706_list !! (n-1)

%o a258706_list = f a000040_list where

%o f ps'@(p:ps) | any (== 0) (map a010051' dps) = f ps

%o | otherwise = p : f (ps' \\ dps)

%o where dps = map read $ permutations $ show p

%o -- _Reinhard Zumkeller_, Jun 10 2015

%o (PARI) From _M. F. Hasler_, Jun 26 2018: (Start)

%o {A=[2,5]; for(n=1, 317, my(D=[1,3,7,9], r=10^n\9); for(a=1,4, for(b=a^(n<3),4, for(j=0, if(b!=a,n-1), ispseudoprime(D[a]*r+(D[b]-D[a])*10^j)||next(2)); A=setunion(A, [r*D[a]+(D[b]-D[a])*10^if(b<a,n-1)])))); A}

%o is(n)={(n=digits(n))[#n]>=n[1] && #select(d->d,n[^1]-n[^-1])<2 && !for(i=1,(#n)^(n[#n]>1), isprime(fromdigits(n=concat(n[^1],n[1])))||return)} \\ By Johnson's theorem and minimality required here, the number must be of the form ab...b or a...ab (=> first difference of digits has at most 1 nonzero component) and then is sufficient to consider rotations of the digits. (End)

%Y Cf. A003459, A004023, A004022 (subsequence of repunit primes).

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, Jun 09 2015

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Last modified August 6 12:33 EDT 2020. Contains 336246 sequences. (Running on oeis4.)