A068654 -id:A068654 - 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, ", ")))

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

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, ", ")))

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))))

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

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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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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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