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A293657 Base-4 circular primes that are not base-4 repunits. 7
7, 13, 23, 29, 53, 383, 509, 863, 983, 1013 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: The sequence is finite, with 1013 being the last term (see A293142).

Written in base 4 (A007090), the terms are 13, 31, 113, 131, 311, 11333, 13331, 31133, 33113, 33311. - Antti Karttunen, Nov 26 2017

From Michael De Vlieger, Dec 30 2017: (Start)

The digits of primes in this sequence must be in the reduced residue system modulo 4, i.e., {1, 3}.

a(11), if it exists, must be larger than 4^21 = 4398046511104. (End)

LINKS

Table of n, a(n) for n=1..10.

EXAMPLE

53 written in base 4 is 311. The base-4 numbers 311, 131, 113 written in base 10 are 53, 29, 23, respectively and all those numbers are prime, so 23, 29 and 53 are terms of the sequence.

MATHEMATICA

With[{b = 4}, Select[Array[Map[If[Union@ # == {1}, 0, FromDigits[#, b]] &, NestList[RotateLeft, #, Length@ # - 1]] &@ IntegerDigits[Prime@ #, b] &, 10^6, If[PrimeQ@ b, #, # + 1] &@ PrimePi@ b], AllTrue[#, PrimeQ] &][[All, 1]] ] (* Michael De Vlieger, Nov 26 2017 *)

With[{b = 4}, Select[Flatten@ Array[FromDigits[#, b] & /@ Most@ Rest@ Tuples[{1, 3}, #] &, 18, 2], Function[w, And[ AllTrue[ Array[ FromDigits[ RotateRight[w, #], b] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 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, 4))!=vecmax(digits(p, 4)), if(is_circularprime(p, 4), print1(p, ", "))))

CROSSREFS

Cf. A007090, A293142.

Cf. base-b nonrepunit circular primes: A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9), A293663 (b=10).

Sequence in context: A214794 A043104 A106349 * A048449 A147812 A043884

Adjacent sequences:  A293654 A293655 A293656 * A293658 A293659 A293660

KEYWORD

nonn,base,more

AUTHOR

Felix Fröhlich, Oct 28 2017

STATUS

approved

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Last modified May 20 02:48 EDT 2019. Contains 323412 sequences. (Running on oeis4.)