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A100607 Concatenated primes of order 3.

%I A100607 #13 Apr 28 2025 14:43:17
%S A100607 223,227,233,257,277,337,353,373,523,557,577,727,733,757,773,1123,
%T A100607 1153,1327,1373,1723,1733,1753,1777,1933,1973,2113,2137,2213,2237,
%U A100607 2243,2267,2273,2293,2297,2311,2333,2341,2347,2357,2371,2377,2383,2389,2417,2437
%N A100607 Concatenated primes of order 3.
%C A100607 This is a subset of all concatenated primes (A019549). Some of these primes have dual order - example 223. It can be viewed as order two(2 and 23) or as order three (2,2 and 3).
%C A100607 There are 15 such numbers less than 1000 and 202 less than 10^4. - _Robert G. Wilson v_, Dec 03 2004
%H A100607 Robert Israel, <a href="/A100607/b100607.txt">Table of n, a(n) for n = 1..10000</a>
%H A100607 Chris Caldwell, <a href="http://www.utm.edu/research/primes/lists/small/1000.txt">The First thousand primes</a>.
%F A100607 Each of the listed primes is made from three primes (same or different).
%e A100607 257 is in the sequence since it is made from three (distinct) primes.
%p A100607 filter:= proc(n) local m, i, j, ni, nj, np, n3;
%p A100607  if not isprime(n) then return false fi;
%p A100607  m:= ilog10(n);
%p A100607  for i from 1 to m-1 do
%p A100607    ni:= n mod 10^i;
%p A100607    if ni < 10^(i-1) or not isprime(ni) then next fi;
%p A100607    np:= (n-ni)/10^i;
%p A100607    for j from 1 to m-i do
%p A100607      nj:= np mod 10^j;
%p A100607      if nj < 10^(j-1) then next fi;
%p A100607      n3:= (np-nj)/10^j;
%p A100607      if  isprime(nj) and isprime(n3) then return true fi;
%p A100607  od od;
%p A100607  false
%p A100607 end proc;
%p A100607 select(filter, [seq(i,i=3..10000,2)]); # _Robert Israel_, Apr 28 2025
%t A100607 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) t = Sort[ KSubsets[ Flatten[ Table[ Prime[ Range[25]], {3}]], 3]]; lst = {}; Do[k = 1; u = Permutations[t[[n]]]; While[k < Length[u], v = FromDigits[ Flatten[ IntegerDigits /@ u[[k]]]]; If[ PrimeQ[v], AppendTo[lst, v]]; k++ ], {n, Length[t]}]; Take[ Union[lst], 45] (* _Robert G. Wilson v_, Dec 03 2004 *)
%Y A100607 Cf. A019549, A100633, A383195.
%K A100607 easy,nonn,base
%O A100607 1,1
%A A100607 _Parthasarathy Nambi_, Nov 30 2004
%E A100607 Corrected and extended by _Robert G. Wilson v_, Dec 03 2004