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.

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A204844 Cyclic primes that are not absolute primes (A003459).

Original entry on oeis.org

197, 719, 971, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331
Offset: 1

Views

Author

N. J. A. Sloane, Jan 19 2012

Keywords

Comments

Every cyclic permutation of the digits is a prime, but there exists a non-cyclic permutation of the digits that produces a composite. [Extended by Felix Fröhlich, Aug 05 2018]
The sequence is the relative complement of A317688 in A293663. - Felix Fröhlich, Aug 05 2018
Conjecture: The sequence is finite, with 999331 being the last term (cf. A293142). - Felix Fröhlich, Aug 05 2018

Examples

			197, 719 and 971 are terms of the sequence, because all three numbers are prime, each number can be obtained by cyclically permuting the digits of one of the other numbers and there exist some composites, namely 791 and 917, that can be obtained from non-cyclic permutations of the digits of those three numbers. - _Felix Fröhlich_, Aug 10 2018

Links

  • J. L. Boal and J. H. Bevis, Permutable primes, Mathematics Magazine, Vol. 55, No. 1 (1982), 38-41.

Crossrefs

Cf. A003459, A068652, A293142, A293663, A317688.

Programs

  • Mathematica
    Select[Prime@ Range@ PrimePi[10^6], Union[d = IntegerDigits[#], {1,3,7,9}] == {1, 3, 7, 9} && AllTrue[ RotateLeft[d, #] & /@ Range@ IntegerLength@ #, PrimeQ@ FromDigits@ # &] && AnyTrue[ FromDigits /@ Permutations[d], CompositeQ] &] (* Giovanni Resta, Aug 10 2018 *)
  • 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
is_circularprime(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))))
find_index_a(vec) = my(r=#vec-1); while(1, if(vec[r] < vec[r+1], return(r)); r--; if(r==0, return(-1)))
find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1)))
switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec
reverse_order(vec, r) = my(v=[], w=[]); for(x=1, r, v=concat(v, vec[x])); for(y=r+1, #vec, w=concat(w, vec[y])); w=Vecrev(w); concat(v, w)
next_permutation(vec) = my(r=find_index_a(vec)); if(r==-1, return(0), my(s=find_index_b(r, vec)); vec=switch_elements(vec, r, s); vec=reverse_order(vec, r)); vec
is_permutable_prime(n) = if(n < 10, return(1)); my(d=vecsort(digits(n))); while(1, if(!ispseudoprime(eva(d)), return(0)); d=next_permutation(d); if(d==0, return(1)))
forprime(p=1, , if(is_circularprime(p) && !is_permutable_prime(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 05 2018

Extensions

More terms from Felix Fröhlich, Aug 05 2018

A328040 a(n) is the number of integers b with 1 < b < p such that p = prime(n) is a base-b nonrepunit circular prime with at least two base-b digits.

Original entry on oeis.org

0, 0, 1, 3, 4, 7, 9, 7, 11, 12, 15, 14, 18, 23, 20, 28, 18, 24, 30, 31, 35, 34, 32, 29, 48, 41, 40, 45, 35, 54, 58, 50, 56, 54, 47, 43, 78, 47, 74, 70, 50, 69, 63, 93, 82, 78, 78, 103, 69, 62, 82, 79, 82, 87, 68, 92, 100, 80, 120, 89, 117, 91, 112, 132, 97, 93
Offset: 1

Views

Author

Felix Fröhlich, Oct 03 2019

Keywords

Comments

Conjecture: a(n) > 0 for n > 2.

Examples

			For n = 4: 7 is the 4th prime and in base 3, 7 is 21, with 12 equal to 5 in decimal, which is prime, in base 4, 7 is 13, with 31 equal to 13 in decimal, which is prime and in base 5, 7 is 12, with 21 equal to 11 in decimal, which is prime. Altogether, there are 3 such bases, so a(4) = 3.

Crossrefs

Cf. A293142.

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
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
count_bases(n) = my(i=0); for(b=3, n-1, if(vecmin(digits(n, b))!=vecmax(digits(n, b)), if(is_circularprime(n, b), i++))); i
forprime(p=1, 400, print1(count_bases(p), ", "))
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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