A293662 Base-9 circular primes that are not base-9 repunits.

11, 13, 17, 19, 23, 37, 43, 47, 67, 71, 73, 79, 101, 149, 173, 181, 211, 233, 347, 421, 443, 613, 641, 647, 673, 719, 727, 971, 1123, 1361, 1429, 1609, 1697, 2153, 2179, 3371, 3547, 3833, 4019, 4091, 4099, 4229, 5227, 5261, 5281, 5683, 5689, 5741, 5749, 5821

Offset: 1

Author

Felix Fröhlich, Dec 30 2017

Keywords

• nonn
• base

Comments

Conjecture: The sequence is finite.

Examples

			101 written in base 9 is 122. The base-9 numbers 122, 221, 212 written in base 10 are 101, 181, 173, respectively and all those numbers are prime, so 101, 173 and 181 are terms of the sequence.
		

Crossrefs

Cf. A007095, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293663 (b=10).

Programs

Mathematica

With[{b = 9}, Select[Prime@ Range[PrimePi@ b + 1, 10^3], Function[w, And[AllTrue[Array[FromDigits[RotateRight[w, #], b] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 2017 *)

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, 9))!=vecmax(digits(p, 9)), if(is_circularprime(p, 9), print1(p, ", "))))