A317716 -id:A317716 - cp's OEIS frontend

A344626 Primes p such that exactly two numbers among all circular permutations of the digits of p are prime.

13, 17, 31, 37, 71, 73, 79, 97, 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, 937, 941, 947, 977

Offset: 1

Felix Fröhlich, May 25 2021

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

Cf. A270083. Row 2 of A317716.

Cf. primes where exactly k numbers among all circular permutations of digits are prime: A068654 (k=1), A344627 (k=3), A344628 (k=4), A344629 (k=5), A344630 (k=6), A344631 (k=7), A344632 (k=8).

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==2, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A344627 Primes p such that exactly three numbers among all circular permutations of the digits of p are prime.

Original entry on oeis.org

113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1031, 1091, 1097, 1103, 1109, 1123, 1181, 1213, 1231, 1279, 1297, 1301, 1319, 1327, 1579, 1777, 1811, 1873, 1913, 1949, 1951, 1979, 1987, 1993, 2131, 2311, 2377, 2399, 2713, 2791, 2939, 2971, 3011

Offset: 1

Felix Fröhlich, May 25 2021

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

Cf. A270083. Row 3 of A317716.

Cf. primes where exactly k numbers among all circular permutations of digits are prime: A068654 (k=1), A344626 (k=2), A344628 (k=4), A344629 (k=5), A344630 (k=6), A344631 (k=7), A344632 (k=8).

Select[Prime[Range[500]],Total[Boole[PrimeQ[FromDigits/@ Table[ RotateRight[ IntegerDigits[#],n],{n,IntegerLength[#]}]]]]==3&] (* Harvey P. Dale, Mar 30 2023 *)

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==3, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A344628 Primes p such that exactly four numbers among all circular permutations of the digits of p are prime.

Original entry on oeis.org

1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11393, 11701, 11717, 11743, 13177, 13931, 13997, 16993, 17011, 17117, 17431, 17539, 17713, 19717, 19997, 21737, 23339, 23773, 30197, 31139, 31699, 31771, 32377, 33923, 37217, 38197, 39233, 39499, 39799, 39971

Offset: 1

Felix Fröhlich, May 25 2021

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

Cf. A270083. Row 4 of A317716.

Cf. primes where exactly k numbers among all circular permutations of digits are prime: A068654 (k=1), A344626 (k=2), A344627 (k=3), A344629 (k=5), A344630 (k=6), A344631 (k=7), A344632 (k=8).

Select[Prime[Range[4500]],Count[FromDigits/@Table[RotateRight[IntegerDigits[#],d],{d,IntegerLength[ #]}],?PrimeQ]==4&] (* _Harvey P. Dale, Aug 31 2024 *)

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==4, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A344629 Primes p such that exactly five numbers among all circular permutations of the digits of p are prime.

Original entry on oeis.org

11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 103391, 103997, 107119, 110339, 111893, 111919, 113123, 113177, 113983, 114997, 117133, 117319, 117353, 117701, 118931, 119107, 119179, 119191, 119699, 123113, 127733, 129919, 131231, 131771

Offset: 1

Felix Fröhlich, May 25 2021

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

Cf. A270083. Row 5 of A317716.

Cf. primes where exactly k numbers among all circular permutations of digits are prime: A068654 (k=1), A344626 (k=2), A344627 (k=3), A344628 (k=4), A344630 (k=6), A344631 (k=7), A344632 (k=8).

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==5, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A344630 Primes p such that exactly six numbers among all circular permutations of the digits of p are prime.

Original entry on oeis.org

193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331, 1313999, 1317727, 1399913, 1731893, 1743737, 1772713, 1893173, 1977779, 2713177, 3139991, 3173189, 3177271, 3189317, 3717437, 4373717, 7174373, 7271317, 7318931

Offset: 1

Felix Fröhlich, May 25 2021

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

Cf. A270083. Row 6 of A317716.

Cf. primes where exactly k numbers among all circular permutations of digits are prime: A068654 (k=1), A344626 (k=2), A344627 (k=3), A344628 (k=4), A344629 (k=5), A344631 (k=7), A344632 (k=8).

Select[Prime[Range[500000]],Total[Boole[PrimeQ[FromDigits/@Table[RotateRight[IntegerDigits[#],n],{n,0,IntegerLength[ #]-1}]]]]==6&] (* Harvey P. Dale, Sep 22 2024 *)

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==6, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A344631 Primes p such that exactly seven numbers among all circular permutations of the digits of p are prime.

Original entry on oeis.org

17773937, 39371777, 71777393, 73937177, 77393717, 77739371, 93717773, 101717933, 101793137, 111766999, 111897767, 113379997, 113719261, 113773021, 113913133, 117669991, 118977671, 119307977, 119937137, 123975113, 131239751, 131331139, 131473193, 133113913

Offset: 1

Felix Fröhlich, May 25 2021

Cf. A270083. Row 7 of A317716.

Cf. primes where exactly k numbers among all circular permutations of digits are prime: A068654 (k=1), A344626 (k=2), A344627 (k=3), A344628 (k=4), A344629 (k=5), A344630 (k=6), A344632 (k=8).

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==7, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A344632 Primes p such that exactly eight numbers among all circular permutations of the digits of p are prime.

Original entry on oeis.org

119139133, 133119139, 139133119, 191391331, 311913913, 331191391, 913311913, 913913311, 1013517313, 1033939939, 1039191919, 1112795317, 1113194339, 1117923797, 1127953171, 1131943391, 1139937913, 1173917197, 1179237971, 1279531711, 1310135173, 1311399379

Offset: 1

Felix Fröhlich, May 25 2021

Cf. A270083. Row 8 of A317716.

Cf. primes where exactly k numbers among all circular permutations of digits are prime: A068654 (k=1), A344626 (k=2), A344627 (k=3), A344628 (k=4), A344629 (k=5), A344630 (k=6), A344631 (k=7).

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==8, 1, 0)
forprime(p=1, , if(is(p), print1(p, ", ")))

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)).

cp's OEIS Frontend

A344626 Primes p such that exactly two numbers among all circular permutations of the digits of p are prime.

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A344627 Primes p such that exactly three numbers among all circular permutations of the digits of p are prime.

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A344628 Primes p such that exactly four numbers among all circular permutations of the digits of p are prime.

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A344629 Primes p such that exactly five numbers among all circular permutations of the digits of p are prime.

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A344630 Primes p such that exactly six numbers among all circular permutations of the digits of p are prime.

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A344631 Primes p such that exactly seven numbers among all circular permutations of the digits of p are prime.

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A344632 Primes p such that exactly eight numbers among all circular permutations of the digits of p are prime.

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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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