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A317756 Number of distinct primes obtained by cyclically shifting the decimal digits of the n-th prime.

%I A317756 #21 Jan 18 2025 19:04:09
%S A317756 1,1,1,1,1,2,2,1,1,1,2,2,1,1,1,1,1,1,1,2,2,2,1,1,2,2,2,2,1,3,2,3,1,1,
%T A317756 2,1,2,2,1,2,1,2,2,1,3,3,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,2,3,2,2,2,3,
%U A317756 1,1,1,2,2,3,2,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,2
%N A317756 Number of distinct primes obtained by cyclically shifting the decimal digits of the n-th prime.
%C A317756 First occurrence of k, k=1,2,3,...: 2, 13, 113, 1193, 11939, 193939, 17773937, 119139133, ..., . A247153.
%C A317756 a(n) is equal to the row index of prime(n) in A317716.
%C A317756 Every positive integer occurs in this sequence if and only if A247153(i) != 0 for every i >= 1.
%H A317756 Robert Israel, <a href="/A317756/b317756.txt">Table of n, a(n) for n = 1..10000</a>
%F A317756 a(n) = A262988(A000040(n)).
%t A317756 f[n_] := Block[{len = IntegerLength@n, s = {n}}, Do[AppendTo[s, FromDigits@RotateRight@IntegerDigits@s[[k - 1]]], {k, 2, len}]; DeleteDuplicates@Select[s, PrimeQ]] (* after Michael De Vlieger in A262988 *); Array[Length@f@Prime@# &, 105] (* _Robert G. Wilson v_, Aug 06 2018 *)
%t A317756 Table[Count[Union[FromDigits/@Table[RotateRight[IntegerDigits[p],n],{n,IntegerLength[p]}]],_?PrimeQ],{p,Prime[Range[120]]}] (* _Harvey P. Dale_, Jan 18 2025 *)
%o A317756 (PARI) eva(n) = subst(Pol(n), x, 10)
%o A317756 rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
%o A317756 count_primes(n) = my(d=digits(n), i=0); while(1, if(ispseudoprime(eva(d)), i++); d=rot(d); if(d==digits(n), return(i)))
%o A317756 a(n) = my(p=prime(n)); count_primes(p) \\ _Felix Fröhlich_, Aug 06 2018
%Y A317756 Cf. A000040, A262988, A247153, A317716.
%K A317756 base,easy,nonn
%O A317756 1,6
%A A317756 _Felix Fröhlich_ and _Robert G. Wilson v_, Aug 06 2018