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Primes with embedded primes.
1

%I #19 Nov 12 2023 22:17:43

%S 13,17,23,29,31,37,43,47,53,59,67,71,73,79,83,97,103,107,113,127,131,

%T 137,139,151,157,163,167,173,179,191,193,197,199,211,223,227,229,233,

%U 239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337

%N Primes with embedded primes.

%C A079066(a(n)) > 0. - _Reinhard Zumkeller_, Jul 19 2011

%C Is there a prime such that the previous prime is embedded in it? - _Ivan N. Ianakiev_, Nov 09 2023. Answer from _Amiram Eldar_: No. If prime(n) is embedded in prime(n+1) then prime(n+1) has at least one digit more than prime(n), so prime(n+1) > 2 * prime(n). But according to Bertrand's postulate, prime(n+1) < 2*prime(n).

%H Reinhard Zumkeller, <a href="/A179924/b179924.txt">Table of n, a(n) for n = 1..10000</a>

%t f[n_] := Block[{id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[Partition[id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 68, f@# > 1 &]

%o (Haskell)

%o import Data.List (findIndices)

%o a179924 n = a179924_list !! (n-1)

%o a179924_list = map (a000040 . (+ 1)) $ findIndices (> 0) a079066_list

%o -- _Reinhard Zumkeller_, Jul 19 2011

%Y Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274, A179922.

%K base,easy,nonn

%O 1,1

%A _Robert G. Wilson v_, Aug 01 2010