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

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, 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, 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

Offset: 1

Author

Felix Fröhlich and Robert G. Wilson v, Aug 06 2018

Keywords

• base
• easy
• nonn

Comments

First occurrence of k, k=1,2,3,...: 2, 13, 113, 1193, 11939, 193939, 17773937, 119139133, ..., . A247153.

a(n) is equal to the row index of prime(n) in A317716.

Every positive integer occurs in this sequence if and only if A247153(i) != 0 for every i >= 1.

Links

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

Crossrefs

Cf. A000040, A262988, A247153, A317716.

Programs

Mathematica

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 *)
Table[Count[Union[FromDigits/@Table[RotateRight[IntegerDigits[p],n],{n,IntegerLength[p]}]],?PrimeQ],{p,Prime[Range[120]]}] (* _Harvey P. Dale, Jan 18 2025 *)

PARI

eva(n) = subst(Pol(n), x, 10)
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
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)))
a(n) = my(p=prime(n)); count_primes(p) \\ Felix Fröhlich, Aug 06 2018

Formula

a(n) = A262988(A000040(n)).