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

43, 47, 97, 101, 151, 157, 167, 199, 281, 293, 487, 601, 607, 809, 839, 967, 1013, 1069, 1129, 1223, 1249, 1259, 1289, 1361, 1367, 1543, 1571, 1663, 1753, 1861, 1871, 1873, 1997, 2141, 2281, 2551, 2593, 2909, 3121, 3271, 3313, 3361, 3371, 3461, 3823, 3881, 3907, 4019, 4211, 4289, 4327, 4349, 4451, 4513

Offset: 1

Author

Will Gosnell and Robert Israel, Jun 05 2025

Keywords

• nonn
• base

Comments

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014612, A383114, A384636.

Programs

Maple

cat3:= proc(a,b,c)
(a*10^(1+ilog10(b))+b)*10^(1+ilog10(c))+c
end proc;
R:= NULL: count:= 0: a:= 2: b:= 3: c:= 5:
for i from 1 while count < 100 do
  a:= b; b:= c; c:= nextprime(c);
  if numtheory:-bigomega(cat3(a,b,c)) = 3 and numtheory:-bigomega(cat3(c,b,a)) = 3 then
     R:= R,a; count:= count+1;
  fi
od:
R;

Mathematica

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 *)