A286333 Primes p where all the cyclic shifts of their digits to the left also produce primes except the last one before reaching p again.

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 101, 103, 107, 127, 149, 157, 163, 181, 191, 307, 317, 331, 359, 367, 701, 709, 727, 739, 757, 761, 787, 797, 907, 937, 941, 947, 983, 1103, 1109, 1123, 1181, 1301, 1319, 1327, 1949, 1951, 1979, 1987, 1993, 3121, 3187, 3361, 3373, 3701

Offset: 1

Author

Mikk Heidemaa, May 07 2017

Keywords

• nonn
• base

Comments

a(144) = 733793111393, a(145) is larger than 10^16 (if it exists).

Can be considered as primitive terms of A270083, i.e. terms in A270083 can be obtained by cyclic shifts of the digits of terms in this sequence (and leading zeros are not allowed). - Chai Wah Wu, May 21 2017

Examples

			1123 is a member as all the cyclic shifts of its digits to the left result are primes (1231, 2311) except the last one (3112) before reaching the original prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..144

Crossrefs

Cf. A270083 (subsequence of), A286415.

Programs

Mathematica

cyclDigs[k_]:= FromDigits/@ NestList[RotateLeft, IntegerDigits[k], IntegerLength[k]-1]; lftSftNearCircPrmsInBtw[m_, n_]:= ParallelMap[If[ AllTrue[Most[cyclDigs[#]], PrimeQ] && Not@ PrimeQ[Last[cyclDigs[#]]], #, Nothing] &, Prime @ Range[PrimePi[m], PrimePi[n]]];
lftSftNearCircPrmsInBtw[19, 10^7]

Python

from itertools import product
from sympy import isprime
A286333_list = []
for l in range(14):
    for w in product('1379',repeat=l):
        for d in '0123456789':
            for t in '1379':
                s = ''.join(w)+d+t
                n = int(s)
                for i in range(l+1):
                    if not isprime(int(s)):
                        break
                    s = s[1:]+s[0]
                else:
                    if n > 10 and not isprime(int(s)):
                        A286333_list.append(n) # Chai Wah Wu, May 21 2017