A293658 Base-5 circular primes that are not base-5 repunits.

7, 11, 13, 17, 19, 23, 167, 211, 239, 283, 359, 431, 547, 571, 1069, 1249, 1733, 2221, 2417, 2713, 2749, 3049, 3109, 3121

Offset

1,1

Comments

Conjecture: The sequence is finite, with 3121 being the last term (see A293142).

Written in base 5 (A007091), the terms are 12, 21, 23, 32, 34, 43, 1132, 1321, 1424, 2113, 2414, 3211, 4142, 4241, 13234, 14444, 23413, 32341, 34132, 41323, 41444, 44144, 44414, 44441. - Antti Karttunen, Nov 26 2017

a(25), if it exists, must be larger than prime(10^6) = 15485863, an 11-digit quinary number. - Michael De Vlieger, Nov 26 2017

Links

Table of n, a(n) for n=1..24.

Example

1069 written in base 5 is 13234. The base-5 numbers 13234, 32341, 23413, 34132, 41323 written in base 10 are 1069, 2221, 1733, 2417, 2713, respectively and all those numbers are prime, so 1069, 1733, 2221, 2417 and 2713 are terms of the sequence.

Mathematica

Select[Array[Map[If[Union@ # == {1}, 0, FromDigits[#, 5]] &, NestList[RotateLeft, #, Length@ # - 1]] &@ IntegerDigits[Prime@ #, 5] &, 10^5, 4], AllTrue[#, PrimeQ] &][[All, 1]] (* Michael De Vlieger, Nov 26 2017 *)

Prog

(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))))
forprime(p=1, , if(vecmin(digits(p, 5))!=vecmax(digits(p, 5)), if(is_circularprime(p, 5), print1(p, ", "))))

Crossrefs

Cf. A007091, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9), A293663 (b=10).

Sequence in context: A038961 A127903 A271000 * A168079 A296928 A358743

Adjacent sequences: A293655 A293656 A293657 * A293659 A293660 A293661

Keyword

nonn,base,more

Author

Felix Fröhlich, Oct 28 2017

Status

approved