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A384638 Primes p such that the concatenations of three consecutive primes starting with p, in both forward and backwards orders, are triprimes.

%I A384638 #29 Jun 20 2025 15:41:36
%S A384638 43,47,97,101,151,157,167,199,281,293,487,601,607,809,839,967,1013,
%T A384638 1069,1129,1223,1249,1259,1289,1361,1367,1543,1571,1663,1753,1861,
%U A384638 1871,1873,1997,2141,2281,2551,2593,2909,3121,3271,3313,3361,3371,3461,3823,3881,3907,4019,4211,4289,4327,4349,4451,4513
%N A384638 Primes p such that the concatenations of three consecutive primes starting with p, in both forward and backwards orders, are triprimes.
%C A384638 Primes p such that if q and r are the next two primes, both concatenations p||q||r and r||q||p have three prime factors, counted with multiplicity.
%H A384638 Robert Israel, <a href="/A384638/b384638.txt">Table of n, a(n) for n = 1..10000</a>
%e A384638 a(2) = 47 is a term because 47, 53, 59 are consecutive primes and both 475359 = 3 * 193 * 821 and 595347 = 3 * 191 * 1039 have three prime factors, counted with multiplicity.
%p A384638 cat3:= proc(a,b,c)
%p A384638 (a*10^(1+ilog10(b))+b)*10^(1+ilog10(c))+c
%p A384638 end proc;
%p A384638 R:= NULL: count:= 0: a:= 2: b:= 3: c:= 5:
%p A384638 for i from 1 while count < 100 do
%p A384638   a:= b; b:= c; c:= nextprime(c);
%p A384638   if numtheory:-bigomega(cat3(a,b,c)) = 3 and numtheory:-bigomega(cat3(c,b,a)) = 3 then
%p A384638      R:= R,a; count:= count+1;
%p A384638   fi
%p A384638 od:
%p A384638 R;
%t A384638 Select[Prime[Range[612]],PrimeOmega[FromDigits[Join[IntegerDigits[#],IntegerDigits[NextPrime[#]],IntegerDigits[NextPrime[#,2]]]]]==3&&PrimeOmega[FromDigits[Join[IntegerDigits[NextPrime[#,2]],IntegerDigits[NextPrime[#,1]],IntegerDigits[#]]]]==3&] (* _James C. McMahon_, Jun 20 2025 *)
%Y A384638 Cf. A014612, A383114, A384636.
%K A384638 nonn,base
%O A384638 1,1
%A A384638 _Will Gosnell_ and _Robert Israel_, Jun 05 2025