A068653 -id:A068653 - cp's OEIS frontend

A068654 Prime numbers such that every cyclic permutation (other than the number itself) is composite.

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 109, 137, 139, 151, 167, 179, 193, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 281, 283, 293, 347, 349, 353, 383, 389, 401, 409, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 487, 499, 503, 509

Offset: 1

Amarnath Murthy, Feb 28 2002

			167 is a member as the two cyclic permutations other than the number itself i.e. 671 and 716 are composite.

Cf. A003459, A068652, A068653.

Select[Prime[Range[100]],Union[PrimeQ[FromDigits/@Table[ RotateRight[ IntegerDigits[#],i],{i,IntegerLength[#]-1}]]]=={False}&] (* Harvey P. Dale, Dec 08 2012 *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 21 2002

A262988 Number of distinct primes, including n if prime, obtained by cyclically shifting the digits of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Felix Fröhlich, Oct 06 2015

First differs from A039999 at n = 103.
Differs from A061264 iff n is a term of A004022.
a(n) = A055642(n) iff n is a term of A068652, except when n is also in A004022.

			a(1013) = 2, because of the four cyclic permutations of the digits of 1013 (1013, 131, 1310, 3101) two, namely 1013 and 131, are prime and those two primes are distinct.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A016114, A039999, A045978, A061264, A068652, A068653, A068654, A069219, A247153.

f[n_] := Block[{len = IntegerLength@ n, s = {n}}, Do[AppendTo[s, FromDigits@ RotateRight@ IntegerDigits@ s[[k - 1]]], {k, 2, len}]; DeleteDuplicates@ Select[s, PrimeQ]]; Array[ Length@ f@ # &, {87}] (* Michael De Vlieger, Oct 07 2015 *)

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
eva(n) = x=0; for(k=1, #n, x=x+(n[k]*10^(#n-k))); x
a(n) = i=0; r=rot(digits(n)); while(r!=digits(n), if(ispseudoprime(eva(r)), i++); r=rot(r)); if(ispseudoprime(eva(r)), i++); i

A224400 Composite numbers that become prime when their digits are put in nondecreasing order.

Original entry on oeis.org

20, 30, 32, 50, 70, 74, 76, 91, 92, 95, 98, 110, 130, 170, 172, 175, 176, 190, 194, 200, 203, 209, 217, 230, 232, 272, 275, 290, 292, 296, 300, 301, 302, 310, 319, 320, 322, 323, 329, 332, 370, 371, 374, 376, 391, 392, 394, 395, 398, 407, 437, 470, 473, 475

Offset: 1

Christian N. K. Anderson and Kevin L. Schwartz, Apr 05 2013

Conjecture: a(n) ~ 7.75*n.

			217 = 7*31, but 127 is prime. 302 = 2*151, but 23 is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Subset of A007935.

Cf. A068653, A076055, A224402.

Select[Range[500], ! PrimeQ[#] && PrimeQ[FromDigits[Sort[IntegerDigits[#]]]] &] (* T. D. Noe, Apr 05 2013 *)

j=1; y=c(); library(gmp)
while(length(y)<1000) {
if(isprime((j=j+1))==0) {
x=sort(as.numeric(strsplit(as.character(j),spl="")[[1]]))
if(isprime(paste(x[x>0],collapse=""))>0) y=c(y,j) #drop leading zeros
}
}

A224402 Composite numbers that become prime when their digits are put in nonincreasing order.

Original entry on oeis.org

14, 16, 34, 35, 38, 112, 118, 119, 121, 124, 125, 128, 133, 134, 136, 142, 143, 145, 146, 152, 154, 164, 166, 175, 176, 182, 188, 194, 214, 215, 218, 314, 316, 334, 341, 343, 344, 346, 356, 358, 361, 364, 365, 368, 374, 377, 385, 386, 388, 395, 398, 412, 413

Offset: 1

Christian N. K. Anderson and Kevin L. Schwartz, Apr 05 2013

Because any number ending in zero is composite, the sequence experiences gaps of at least order O(a(n))-1 between changes in the most significant digit.

			194=2*97, but 941 is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Subset of A007935.

Cf. A068653, A076055, A224400.

Select[Range[500], ! PrimeQ[#] && PrimeQ[FromDigits[Reverse[Sort[IntegerDigits[#]]]]] &] (* T. D. Noe, Apr 05 2013 *)

j=1; y=c(); library(gmp)
while(length(y)<1000) {
if(isprime((j=j+1))==0) {
x=sort(as.numeric(strsplit(as.character(j),spl="")[[1]]),decr=T)
if(isprime(paste(x,collapse=""))>0) y=c(y,j)
}
}

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 }.

A225193 Composite numbers such that every non-identity permutation gives a prime.

Original entry on oeis.org

14, 16, 20, 30, 32, 34, 35, 38, 50, 70, 74, 76, 91, 92, 95, 98, 110, 118, 119, 133, 772, 775, 778, 779, 1118, 3337, 7771, 77779

Offset: 1

Jayanta Basu, May 01 2013

			772 is a member since both 727 and 277 are primes.

Cf. A003459, A068652, A068653.

t={}; Do[p=Permutations[IntegerDigits[n]]; c=Length[p]; cn=Length[Select[Table[FromDigits[k],{k,p}], PrimeQ]]; If[!PrimeQ[n] && c>1 && cn==c-1, AppendTo[t,n]], {n,10,100000}]; t

from sympy import isprime
from itertools import count, islice, permutations
def agen(): yield from (k for k in count(1) if len(set(s:=str(k)))!=1 and not isprime(k) and all((t:=int("".join(m)))==k or isprime(t) for m in permutations(s)))
print(list(islice(agen(), 28))) # Michael S. Branicky, Dec 29 2023

cp's OEIS Frontend

A068654 Prime numbers such that every cyclic permutation (other than the number itself) is composite.

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A262988 Number of distinct primes, including n if prime, obtained by cyclically shifting the digits of n.

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A224400 Composite numbers that become prime when their digits are put in nondecreasing order.

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A224402 Composite numbers that become prime when their digits are put in nonincreasing order.

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R

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

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A225193 Composite numbers such that every non-identity permutation gives a prime.

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