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A155032 Primes p such that both pi(p) and the concatenation of pi(p) and p are prime, where pi is the prime counting function.

%I A155032 #16 Jul 20 2025 20:00:49
%S A155032 3,59,83,179,283,353,431,709,1433,2269,2381,3559,3593,4153,5503,6899,
%T A155032 7109,7351,7649,8513,11909,13297,14107,14437,14591,16073,16127,16451,
%U A155032 16901,17117,17539,17987,18149,19777,20759,21317,22027,24439,25357,26783,27437,29269,30253,32299,34057,34259,34421,34543,35617,36307,37049
%N A155032 Primes p such that both pi(p) and the concatenation of pi(p) and p are prime, where pi is the prime counting function.
%H A155032 Harvey P. Dale, <a href="/A155032/b155032.txt">Table of n, a(n) for n = 1..1000</a>
%e A155032 Since 3 is the second prime number and the concatenation of 2 and 3 gives 23, which is prime, 3 is in the sequence.
%e A155032 Since 59 is the seventeenth prime and the concatenation of 17 and 59 gives 1759, another prime, 59 is also in the sequence.
%t A155032 (* First run the program given for A154963 *) Prime[A154963]
%t A155032 Module[{nn=1000},Select[Table[{n,Prime[n]},{n,Prime[Range[nn]]}],PrimeQ[#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]]&]][[;;,2]] (* _Harvey P. Dale_, Jul 20 2025 *)
%Y A155032 pi(a(n)) = A154963(n).
%K A155032 nonn,base
%O A155032 1,1
%A A155032 _Juri-Stepan Gerasimov_, Jan 19 2009
%E A155032 Edited and extended beyond a(3) by _Alonso del Arte_, Jan 20 2009, with thanks to _Klaus Brockhaus_'s edit of A154963
%E A155032 Name clarified by _Sean A. Irvine_, Jul 20 2025