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.
3, 59, 83, 179, 283, 353, 431, 709, 1433, 2269, 2381, 3559, 3593, 4153, 5503, 6899, 7109, 7351, 7649, 8513, 11909, 13297, 14107, 14437, 14591, 16073, 16127, 16451, 16901, 17117, 17539, 17987, 18149, 19777, 20759, 21317, 22027, 24439, 25357, 26783, 27437, 29269, 30253, 32299, 34057, 34259, 34421, 34543, 35617, 36307, 37049
Offset: 1
Examples
Since 3 is the second prime number and the concatenation of 2 and 3 gives 23, which is prime, 3 is in the sequence. Since 59 is the seventeenth prime and the concatenation of 17 and 59 gives 1759, another prime, 59 is also in the sequence.
Links
- Harvey P. Dale, Table of n, a(n) for n = 1..1000
Crossrefs
pi(a(n)) = A154963(n).
Programs
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Mathematica
(* First run the program given for A154963 *) Prime[A154963] 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 *)
Extensions
Edited and extended beyond a(3) by Alonso del Arte, Jan 20 2009, with thanks to Klaus Brockhaus's edit of A154963
Name clarified by Sean A. Irvine, Jul 20 2025