A380655 Smallest prime p > 10^(n-1) for which successive cyclic shifts of digits by 1, ..., n-1 positions to the left are all prime, or -1 if no such p exists.

2, 11, 113, 1193, 11939, 193939, 71777393, 913311913, 93739179151, 317793117877, 731779311787, 1373779119729007

Offset: 1

Author

Jean-Marc Rebert, Jan 29 2025

Keywords

• nonn
• base
• more

Examples

			_n_p__shifts of digits by 1, ..., n-1 positions (n <= number of digits of p) to the left
 1 2 -> ;
 2 11 -> 11;
 3 113 -> 131, 311;
 4 1193 -> 1931, 9311, 3119;
 5 11939 -> 19391, 93911, 39119, 91193;
 6 193939 -> 939391, 393919, 939193, 391939, 919393;
 7 71777393 -> 17773937, 77739371, 77393717, 73937177, 39371777, 93717773, but 37177739 = 29 * 683 * 1877;
 8 913311913 -> 133119139, 331191391, 311913913, 119139133, 191391331, 913913311, 139133119, but 391331191 = 29 * 131 * 239 * 431;
 9 93739179151 -> 37391791519, 73917915193, 39179151937, 91791519373, 17915193739, 79151937391, 91519373917, 15193739179, but 51937391791 = 419 * 887 * 139747;
10 317793117877 -> 177931178773, 779311787731, 793117877317, 931178773177, 311787731779, 117877317793, 178773177931, 787731779311, 877317793117, but 773177931178 = 2 * 386588965589;
11 731779311787 -> 317793117877, 177931178773, 779311787731, 793117877317, 931178773177, 311787731779, 117877317793, 178773177931, 787731779311, 877317793117, but 773177931178 = 2 * 386588965589;
12 1373779119729007 -> 3737791197290071, 7377911972900713, 3779119729007137, 7791197290071373, 7911972900713737, 9119729007137377, 1197290071373779, 1972900713737791, 9729007137377911, 7290071373779119, 2900713737791197, but 9007137377911972 = 2^2 * 13 * 6841 * 25320008821;
		

Crossrefs

Cf. A247153, A380669.

Programs

Python

from itertools import count, product
from sympy import isprime
def A380655(n):
    if n == 1: return 2
    for l in count(n):
        for a in product('1379', repeat=n-1):
            for b in product('0123456789', repeat=l-n):
                for c in '1379':
                    d = ''.join(a+b)+c
                    if all(isprime(int(d[i:]+d[:i])) for i in range(n)):
                        return int(d) # Chai Wah Wu, Jan 30 2025

Extensions

a(10) and a(11) corrected by Chai Wah Wu, Jan 30 2025

Name edited by Pontus von Brömssen, Feb 03 2025

a(12) from Chai Wah Wu, Feb 06 2025