A262988 -id:A262988 - cp's OEIS frontend

A270083 Near-miss circular primes: Primes p where all but one of the numbers obtained by cyclically permuting the digits of p are prime.

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 101, 103, 107, 127, 149, 157, 163, 173, 181, 191, 271, 277, 307, 313, 317, 331, 359, 367, 379, 397, 419, 479, 491, 571, 577, 593, 617, 631, 673, 701, 709, 727, 739, 757, 761, 787, 797, 811, 839, 877, 907, 911

Offset: 1

Felix Fröhlich, Mar 10 2016

Prime p is a term of the sequence iff A262988(p) = A055642(p) - 1.
If a(512) exists, it is larger than 10^16. - Giovanni Resta, Apr 27 2017
If one of the digits is even or 5, the miss occurs when that digit is permuted to the ones place. Avoiding that simple obstruction, this sequence intersected with A091633 is 19, 173, 191, 313, 317, 331, 379, 397, 739, 797, 911, 937, 977, 1319, 1777, 1913, 1979, 1993, 3191, 3373, 3719, 3733, 3793, ... . Is this an infinite subsequence? - Danny Rorabaugh, May 15 2017

Felix Fröhlich and Giovanni Resta, Table of n, a(n) for n = 1..511 (first 487 terms from Felix Fröhlich)

Cf. A045978, A068652, A262988.

NearCircPrmsUpTo10powerK[k_]:= Union @ Flatten[ Table[ParallelMap[If[(Count[FromDigits /@ NestList[RotateLeft, IntegerDigits[#], IntegerLength[#]-1], ?PrimeQ] == IntegerLength[#]-1), #, Nothing] &, Select[FromDigits /@ Tuples[Range[0, 9], n], PrimeQ]], {n, k}], 1]; NearCircPrmsUpTo10powerK[7] (* _Mikk Heidemaa, Apr 26 2017 *)

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) = subst(Pol(n), x, 10)
is(n) = my(r=rot(digits(n)), i=0); while(r!=digits(n), if(ispseudoprime(eva(r)), i++); r=rot(r)); if(ispseudoprime(eva(r)), i++); if(n==1 || n==11, return(0)); if(i==#Str(n)-1, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A293663 Circular primes that are not repunits.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939

Offset: 1

Felix Fröhlich, Dec 30 2017

Relative complement of A004022 in A068652.
Conjecture: The sequence is finite.
From Michael De Vlieger, Dec 30 2017: (Start)
Primes > 5 in this sequence must only have digits that are in the reduced residue system modulo 10, i.e., {1, 3, 7, 9}.
There are 54 terms that have 6 or fewer decimal digits, the largest of which is 999331.
a(55) must be larger than 10^11. (End) [Corrected by Felix Fröhlich, Mar 15 + 24 2019]
From Felix Fröhlich, Mar 16 2019: (Start)
a(55) > 10^23 if it exists (cf. De Geest link).
Numbers k such that A262988(k) = A055642(k). (End)

			The numbers resulting from cyclic permutations of the digits of 1193 are 1931, 9311 and 3119, respectively and all those numbers are prime, so 1193, 1931, 3119 and 9311 are terms of the sequence.

Felix Fröhlich, Table of n, a(n) for n = 1..54
P. De Geest, Circular Primes

Cf. A004022, A055642, A068652, A262988, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9).

Select[Prime@ Range[10^5], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* or *)
Select[Flatten@ Array[FromDigits /@ Most@ Rest@ Tuples[{1, 3, 7, 9}, #] &, 9, 2], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 30 2017 *)

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_circularprime(p) = my(d=digits(p), 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))))
forprime(p=1, , if(vecmax(digits(p)) > 1, if(is_circularprime(p), print1(p, ", "))))

/* The following is a much faster program that only tests numbers whose decimal expansion consists of digits from the set {1, 3, 7, 9}. */
eva(n) = subst(Pol(n), x, 10)
next_v(vec) = my(k=#vec); if(vecmin(vec)==9, vec=concat(vector(#vec, t, 1), [3]); return(vec)); while(k > 0, if(vec[k]==9, vec[k]=1, if(vec[k]==3, vec[k]=7; return(vec), vec[k]=vec[k]+2, return(vec))); k--)
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
search(n) = my(d=digits(n), e=[], ed=0); while(1, e=rot(d); while(1, if(!ispseudoprime(eva(e)), break, e=rot(e); if(e==d && ispseudoprime(eva(e)), print1(eva(d), ", "); break))); d=next_v(d))
searchfrom(n) = if(n < 12, forprime(p=n, 10, print1(p, ", ")); search(13), my(d=digits(n)); for(k=1, #d, if(d[k]%2==0, d[k]++, if(d[k]==5, d[k]=7))); search(eva(d)))
/* Start a search from 1 upwards as follows: */
searchfrom(1) \\ Felix Fröhlich, Mar 23 2019

A317716 Square array A(n, k), read by antidiagonals downwards: k-th prime p such that cyclic digit shifts produce exactly n different primes.

Original entry on oeis.org

2, 3, 13, 5, 17, 113, 7, 31, 131, 1193, 11, 37, 197, 1931, 11939, 19, 71, 199, 3119, 19391, 193939, 23, 73, 311, 3779, 19937, 199933, 17773937, 29, 79, 337, 7793, 37199, 319993, 39371777, 119139133, 41, 97, 373, 7937, 39119, 331999, 71777393, 133119139

Offset: 1

Felix Fröhlich, Aug 05 2018

k-th prime p such that A262988(p) = n.
Are all rows of the array infinite?
A term q of A270083 occurs in row A055642(q) - 1 in this array.
A term r of A293663 occurs in row A055642(r) in this array.
Row 1 is a supersequence of A004022.
Column 1 is A247153.

			Array starts
          2,         3,         5,         7,        11,        19,        23, ...
         13,        17,        31,        37,        71,        73,        79, ...
        113,       131,       197,       199,       311,       337,       373, ...
       1193,      1931,      3119,      3779,      7793,      7937,      9311, ...
      11939,     19391,     19937,     37199,     39119,     71993,     91193, ...
     193939,    199933,    319993,    331999,    391939,    393919,    919393, ...
   17773937,  39371777,  71777393,  73937177,  77393717,  77739371,  93717773, ...
  119139133, 133119139, 139133119, 191391331, 311913913, 331191391, 913311913, ...
...

Robert G. Wilson v, Antidiagonals n = 1..13, flattened

Cf. A004022, A055642, A247153, A262988, A270083, A293663.

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
count_primes(n) = my(d=digits(n), i=0); while(1, if(ispseudoprime(eva(d)), i++); d=rot(d); if(d==digits(n), return(i)))
row(n, terms) = my(i=0); forprime(p=1, , if(count_primes(p)==n, print1(p, ", "); i++); if(i==terms, print(""); break))
array(rows, cols) = for(x=1, rows, row(x, cols))
array(7, 7) \\ print initial 7 rows and 7 columns of array

A247153 a(n) = smallest prime p for which cyclic digit shifts produce exactly n different primes, or 0 if no such p exists for n.

Original entry on oeis.org

2, 13, 113, 1193, 11939, 193939, 17773937, 119139133, 111133719913, 111119917373, 111393733793, 1117739771979737

Offset: 1

Felix Fröhlich, Nov 21 2014

a(n) is equal to the smallest n-digit non-repunit prime in A016114, unless no n-digit non-repunit prime exists in A016114. In that case, the number of digits of a(n), if it exists, must be > n.
From David A. Corneth, Aug 06 2018: (Start)
Do we have leading digit of a(n) <= any digit from a(n)?
For n > 1, can a(n) contain a digit d with gcd(10, d) > 1? (End)
Smallest prime p such that A262988(p) = n. - Felix Fröhlich, Aug 06 2018

P. De Geest, Circular primes

Cf. A016114, A262988. This is column 1 of A317716.

a(7)-a(8) from P. De Geest's website added by Felix Fröhlich, Nov 26 2014

a(9)-a(12) from Robert G. Wilson v, Aug 06 2018

A317756 Number of distinct primes obtained by cyclically shifting the decimal digits of the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 2, 3, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 2, 2, 2, 3, 1, 1, 1, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Felix Fröhlich and Robert G. Wilson v, Aug 06 2018

First occurrence of k, k=1,2,3,...: 2, 13, 113, 1193, 11939, 193939, 17773937, 119139133, ..., . A247153.
a(n) is equal to the row index of prime(n) in A317716.
Every positive integer occurs in this sequence if and only if A247153(i) != 0 for every i >= 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A262988, A247153, A317716.

f[n_] := Block[{len = IntegerLength@n, s = {n}}, Do[AppendTo[s, FromDigits@RotateRight@IntegerDigits@s[[k - 1]]], {k, 2, len}]; DeleteDuplicates@Select[s, PrimeQ]] (* after Michael De Vlieger in A262988 *); Array[Length@f@Prime@# &, 105] (* Robert G. Wilson v, Aug 06 2018 *)
Table[Count[Union[FromDigits/@Table[RotateRight[IntegerDigits[p],n],{n,IntegerLength[p]}]],?PrimeQ],{p,Prime[Range[120]]}] (* _Harvey P. Dale, Jan 18 2025 *)

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
count_primes(n) = my(d=digits(n), i=0); while(1, if(ispseudoprime(eva(d)), i++); d=rot(d); if(d==digits(n), return(i)))
a(n) = my(p=prime(n)); count_primes(p) \\ Felix Fröhlich, Aug 06 2018

a(n) = A262988(A000040(n)).

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A270083 Near-miss circular primes: Primes p where all but one of the numbers obtained by cyclically permuting the digits of p are prime.

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A293663 Circular primes that are not repunits.

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A317716 Square array A(n, k), read by antidiagonals downwards: k-th prime p such that cyclic digit shifts produce exactly n different primes.

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A247153 a(n) = smallest prime p for which cyclic digit shifts produce exactly n different primes, or 0 if no such p exists for n.

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A317756 Number of distinct primes obtained by cyclically shifting the decimal digits of the n-th prime.

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