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 A293663 Circular primes that are not repunits. 13
 2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Relative complement of A004022 in A068652. Conjecture: The sequence is finite. From Michael De Vlieger, Dec 30 2017: (Start) Primes > 5 in this sequence must only have digits that are in the reduced residue system modulo 10, i.e., {1, 3, 7, 9}. There are 54 terms that have 6 or fewer decimal digits, the largest of which is 999331. a(55) must be larger than 10^11. (End) [Corrected by Felix Fröhlich, Mar 15 + 24 2019] From Felix Fröhlich, Mar 16 2019: (Start) a(55) > 10^23 if it exists (cf. De Geest link). Numbers k such that A262988(k) = A055642(k). (End) LINKS Felix Fröhlich, Table of n, a(n) for n = 1..54 P. De Geest, Circular Primes EXAMPLE The numbers resulting from cyclic permutations of the digits of 1193 are 1931, 9311 and 3119, respectively and all those numbers are prime, so 1193, 1931, 3119 and 9311 are terms of the sequence. MATHEMATICA Select[Prime@ Range[10^5], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* or *) Select[Flatten@ Array[FromDigits /@ Most@ Rest@ Tuples[{1, 3, 7, 9}, #] &, 9, 2], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 30 2017 *) PROG (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(p) = my(d=digits(p), 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)))) forprime(p=1, , if(vecmax(digits(p)) > 1, if(is_circularprime(p), print1(p, ", ")))) (PARI) /* The following is a much faster program that only tests numbers whose decimal expansion consists of digits from the set {1, 3, 7, 9}. */ eva(n) = subst(Pol(n), x, 10) next_v(vec) = my(k=#vec); if(vecmin(vec)==9, vec=concat(vector(#vec, t, 1), [3]); return(vec)); while(k > 0, if(vec[k]==9, vec[k]=1, if(vec[k]==3, vec[k]=7; return(vec), vec[k]=vec[k]+2, return(vec))); k--) 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 search(n) = my(d=digits(n), e=[], ed=0); while(1, e=rot(d); while(1, if(!ispseudoprime(eva(e)), break, e=rot(e); if(e==d && ispseudoprime(eva(e)), print1(eva(d), ", "); break))); d=next_v(d)) searchfrom(n) = if(n < 12, forprime(p=n, 10, print1(p, ", ")); search(13), my(d=digits(n)); for(k=1, #d, if(d[k]%2==0, d[k]++, if(d[k]==5, d[k]=7))); search(eva(d))) /* Start a search from 1 upwards as follows: */ searchfrom(1) \\ Felix Fröhlich, Mar 23 2019 CROSSREFS Cf. A004022, A055642, A068652, A262988, A293142. Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9). Sequence in context: A118724 A055387 A046732 * A317688 A046703 A118722 Adjacent sequences: A293660 A293661 A293662 * A293664 A293665 A293666 KEYWORD nonn,base AUTHOR Felix Fröhlich, Dec 30 2017 STATUS approved

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Last modified September 15 15:25 EDT 2024. Contains 375938 sequences. (Running on oeis4.)