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A293660
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Base-7 circular primes that are not base-7 repunits.
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10
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11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 79, 89, 97, 109, 131, 211, 233, 257, 263, 281, 307, 337, 439, 479, 509, 571, 619, 673, 677, 853, 941, 953, 977, 997, 1021, 1097, 1117, 1163, 1171, 1453, 1511, 1531, 1579, 1597, 1657, 1777, 1787, 1811, 1871, 1933, 1951
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OFFSET
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1,1
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COMMENTS
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Conjecture: The sequence is finite, with 13143449029 being the last term. - [Comment extended by Felix Fröhlich, May 30 2019]
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LINKS
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EXAMPLE
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109 written in base 7 is 214. The base-7 numbers 214, 142, 421 written in base 10 are 109, 79, 211, respectively, and all those numbers are prime, so 79, 109 and 211 are terms of the sequence.
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MATHEMATICA
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With[{b = 7}, Select[Prime@ Range[PrimePi@ b + 1, 300], Function[w, And[AllTrue[Array[FromDigits[RotateRight[w, #], b] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 2017 *)
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PROG
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(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, 7))!=vecmax(digits(p, 7)), if(is_circularprime(p, 7), print1(p, ", "))))
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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