A286415 Primes p where all the cyclic shifts of their digits to the right also produce primes except the last one before reaching p again.

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 173, 271, 277, 313, 379, 397, 419, 479, 491, 571, 577, 593, 617, 631, 673, 811, 839, 877, 911, 977, 1777, 1913, 2131, 2311, 2377, 2399, 2713, 2791, 2939, 2971, 4177, 4339, 4919, 4993, 5119, 5791, 6133, 6737, 6997, 7193, 7333, 7919, 8111

Offset

1,1

Comments

a(125)=937337931113, a(126) is larger than 10^16 (if it exists).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..125

Example

2131 is a member as all the cyclic shifts of its digits to the right result in primes (1213, 3121) except the last one (1312) before reaching the original prime.

Mathematica

cyclDigs[k_]:= FromDigits/@ NestList[RotateRight, IntegerDigits[k], IntegerLength[k]-1]; rgtSftNearCircPrmsInBtw[m_, n_]:= ParallelMap[ If[AllTrue[Most[cyclDigs[#]], PrimeQ] && Not@ PrimeQ[Last[cyclDigs[#]]], #, Nothing] &, Prime @ Range[PrimePi[m], PrimePi[n]]];
rgtSftNearCircPrmsInBtw[19, 10^7]
cspQ[n_]:=Module[{t=PrimeQ[FromDigits/@Table[RotateRight[IntegerDigits[ n], k], {k, IntegerLength[n]-1}]]}, Last[t]==False&&Union[Most[t]]=={True}]; Join[ {19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89}, Select[ Prime[ Range[ 26, 1100]], cspQ]] (* Harvey P. Dale, Oct 05 2020 *)

Prog

(Python)
from itertools import product
from sympy import isprime
A286415_list = []
for l in range(1, 15):
    for d in '123456789':
        for w in product('1379', repeat=l):
            s = d+''.join(w)
            n = int(s)
            for i in range(l):
                if not isprime(int(s)):
                    break
                s = s[-1]+s[:-1]
            else:
                if not isprime(int(s)):
                    A286415_list.append(n) # Chai Wah Wu, May 21 2017

Crossrefs

Cf. A270083 (subsequence of), A286333.

Sequence in context: A286333 A076056 A068654 * A305835 A019384 A088987

Adjacent sequences: A286412 A286413 A286414 * A286416 A286417 A286418

Keyword

nonn,base

Author

Mikk Heidemaa, May 08 2017

Status

approved