cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

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

Views

Author

Felix Fröhlich, Aug 05 2018

Keywords

Comments

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.

Examples

			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, ...
...

Links

Crossrefs

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

Programs

  • PARI
    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

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

Original entry on oeis.org

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

Views

Author

Felix Fröhlich, May 25 2021

Keywords

Links

Crossrefs

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).

Programs

  • PARI
    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

Views

Author

Felix Fröhlich, May 25 2021

Keywords

Links

Crossrefs

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).

Programs

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

Views

Author

Felix Fröhlich, May 25 2021

Keywords

Links

Crossrefs

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).

Programs

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

Views

Author

Felix Fröhlich, May 25 2021

Keywords

Links

Crossrefs

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).

Programs

  • PARI
    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

Views

Author

Felix Fröhlich, May 25 2021

Keywords

Links

Crossrefs

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).

Programs

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

Views

Author

Felix Fröhlich, May 25 2021

Keywords

Crossrefs

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).

Programs

  • PARI
    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

Views

Author

Felix Fröhlich, May 25 2021

Keywords

Crossrefs

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).

Programs

  • PARI
    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, ", ")))

A045978 Palindromic primes that are "near miss circular primes" (all cyclic shifts except one are primes).

Original entry on oeis.org

101, 181, 191, 313, 727, 757, 787, 797, 13931, 71317, 73237, 77977, 7791977
Offset: 0

Views

Author

Carlos Rivera

Keywords

Links

Crossrefs

Cf. A002385, A270083.

Programs

Extensions

Corrected by Jud McCranie, Jan 04 2001

A286415 Primes p where all the cyclic shifts of their digits to the right also produce primes except the last one before reaching p again.

Original entry on oeis.org

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 173, 271, 277, 313, 379, 397, 419, 479, 491, 571, 577, 593, 617, 631, 673, 811, 839, 877, 911, 977, 1777, 1913, 2131, 2311, 2377, 2399, 2713, 2791, 2939, 2971, 4177, 4339, 4919, 4993, 5119, 5791, 6133, 6737, 6997, 7193, 7333, 7919, 8111
Offset: 1

Views

Author

Mikk Heidemaa, May 08 2017

Keywords

Comments

a(125)=937337931113, a(126) is larger than 10^16 (if it exists).

Examples

			2131 is a member as all the cyclic shifts of its digits to the right result in primes (1213, 3121) except the last one (1312) before reaching the original prime.

Links

Crossrefs

Cf. A270083 (subsequence of), A286333.

Programs

  • Mathematica
    cyclDigs[k_]:= FromDigits/@ NestList[RotateRight, IntegerDigits[k], IntegerLength[k]-1]; rgtSftNearCircPrmsInBtw[m_, n_]:= ParallelMap[ If[AllTrue[Most[cyclDigs[#]], PrimeQ] && Not@ PrimeQ[Last[cyclDigs[#]]], #, Nothing] &, Prime @ Range[PrimePi[m], PrimePi[n]]];
rgtSftNearCircPrmsInBtw[19, 10^7]
cspQ[n_]:=Module[{t=PrimeQ[FromDigits/@Table[RotateRight[IntegerDigits[ n],k],{k,IntegerLength[n]-1}]]},Last[t]==False&&Union[Most[t]]=={True}]; Join[ {19,23,29,41,43,47,53,59,61,67,83,89},Select[ Prime[ Range[ 26,1100]],cspQ]] (* Harvey P. Dale, Oct 05 2020 *)
  • Python
    from itertools import product
from sympy import isprime
A286415_list = []
for l in range(1,15):
    for d in '123456789':
        for w in product('1379',repeat=l):
            s = d+''.join(w)
            n = int(s)
            for i in range(l):
                if not isprime(int(s)):
                    break
                s = s[-1]+s[:-1]
            else:
                if not isprime(int(s)):
                    A286415_list.append(n) # Chai Wah Wu, May 21 2017
Showing 1-10 of 13 results. Next

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