A270083 -id:A270083 - cp's OEIS frontend

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

Mikk Heidemaa, May 07 2017

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

			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.

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

Cf. A270083 (subsequence of), A286415.

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]

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

A305885 Zeroless composite numbers which become and remain prime under a complete cyclic shift of digits.

Original entry on oeis.org

14, 16, 32, 34, 35, 38, 74, 76, 91, 92, 95, 98, 118, 119, 133, 176, 194, 316, 398, 712, 715, 731, 736, 772, 775, 778, 779, 793, 794, 914, 935, 973, 1118, 1195, 1312, 1336, 1774, 1937, 3112, 3199, 3337, 3379, 3394, 3772, 3992, 7132, 7198, 7318, 7376, 7771, 7912

Offset: 1

Philip Mizzi, Jun 13 2018

Numbers with a zero digit have been excluded as cycling through these numbers would generate leading zeros, which become problematic throughout the cycle.
3999131, 7919777, 37177739 and 391331191 are in this sequence, see the link. - Eric Chen, Jun 14 2018
The sequence contains all composites without the digit zero that can be obtained by cyclically permuting the digits of the terms of A270083. - Felix Fröhlich, Jul 10 2018

			N_0 = 1118 (composite);
N_1 = 1181 (prime);
N_2 = 1811 (prime);
N_3 = 8111 (prime);
N_4 = N_0 = 1118 (composite).

Robert Israel, Table of n, a(n) for n = 1..121 (first 56 terms from Philip Mizzi)
World of numbers, Circular prime

Cf. A011540, A052382, A068653, A270083.

Q[1]:= [seq([i],i=1..9)]:
for d from 2 to 7 do Q[d]:= map(t -> seq([i,op(t)],i=1..9),Q[d-1]) od:
Res:= NULL: count:= 0:
for d from 2 to 7 do
  for q in Q[d] do P[q]:= isprime(add(q[i]*10^(i-1),i=1..d)) od;
  for q in Q[d] do
     if (not P[q]) and andmap(t -> P[ListTools:-Rotate(q,t)],[$1..d-1]) then
       count:= count+1;
       Res:= Res, add(q[i]*10^(i-1),i=1..d);
     fi
  od
od:
Res; # Robert Israel, Jul 10 2018

Select[Range[11, 8000], Function[{k, d}, And[CompositeQ@ k, FreeQ[d, 0], AllTrue[Array[FromDigits@ RotateLeft[d, #] &, IntegerLength@ k - 1], PrimeQ]]] @@ {#, IntegerDigits@ #} &] (* Michael De Vlieger, Jun 14 2018 *)

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
is(n) = my(d=digits(n), r=rot(d)); if(vecmin(d)==0, return(0), while(1, if(!ispseudoprime(eva(r)), return(0)); r=rot(r); if(r==d, return(1))))
forcomposite(c=1, 8000, if(is(c), print1(c, ", "))) \\ Felix Fröhlich, Jul 10 2018

{ A052382 } intersection { A068653 }.

{ A068653 } minus { A011540 }.

A327822 Numbers k such that when cyclically permuting the digits of k any number of times, any prime obtained is followed by a composite number and vice-versa.

Original entry on oeis.org

14, 16, 19, 20, 23, 29, 30, 32, 34, 35, 38, 41, 43, 47, 50, 53, 59, 61, 67, 70, 74, 76, 83, 89, 91, 92, 95, 98, 1015, 1018, 1070, 1075, 1099, 1132, 1136, 1163, 1216, 1238, 1274, 1303, 1321, 1339, 1361, 1475, 1510, 1517, 1535, 1570, 1574, 1612, 1630, 1631, 1636

Offset: 1

Felix Fröhlich, Sep 26 2019

			When cyclically permuting the digits of 961990 one gets the numbers 961990, 619909, 199096, 990961, 909619, 96199 and these numbers are composite, prime, composite, prime, composite, prime, respectively, so 961990 (and each of these cyclic permutations except 96199) is a term of the sequence.
A more graphical representation:
       961990              C
      /      \           /   \
  096199   619909       P     P
     |        |         |     |
  909619   199096       C     C
      \      /           \   /
       990961              P

Cf. A068652, A068654, A270083.

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
is(n) = my(nn=#Str(n), u=[], v=vector(nn, x, x%2==0), w=vector(nn, x, x%2==1), d=digits(n), r=rot(d)); if(nn%2==1, return(0)); u=concat(u, [ispseudoprime(eva(d))]); u=concat(u, ispseudoprime(eva(r))); while(1, r=rot(r); if(r==d, if(u==v || u==w, return(1)); return(0)); u=concat(u, ispseudoprime(eva(r))))

cp's OEIS Frontend

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.

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A305885 Zeroless composite numbers which become and remain prime under a complete cyclic shift of digits.

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A327822 Numbers k such that when cyclically permuting the digits of k any number of times, any prime obtained is followed by a composite number and vice-versa.

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Author

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Examples

Crossrefs

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PARI