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 A263182 Smallest k such that k//A002275(n)//k is prime, where // denotes concatenation and A002275(n) is the n-th repunit (R_n). 2
 1, 3, 13, 17, 1073, 19, 17, 29, 10000117, 73, 17, 3, 1007, 3, 43, 11, 1000000000000029, 1, 31, 11, 1191, 1, 1143, 31, 10000079, 21, 91, 59, 1019, 3, 67, 117, 10000000000000000000000000000077, 109, 89, 49, 1097, 41, 1053, 43, 10000047, 87, 23, 53, 1149, 83, 57 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) = 3 if n is in A056251. From Chai Wah Wu, Nov 05 2019 : (Start) Theorem: a(2^r*s) >= 10^(2^r-1) for all r >= 0, s > 0. Proof: Note that if k has m digits, then k//A002275(n)//k = k*(10^(n+m)+1) + A002275(n)*10^m which is a multiple of gcd(A002275(n),10^(n+m)+1). Next, since 10^(2^r) - 1 = (10^(2^(r-1) -1))*(10^(2^(r-1) + 1)) and 9 does not divide 10^n+1, by induction it is easy to see that 10^(2^w) + 1 is a divisor of A002275(2^r) for 1 <= w < r. Since A002275(2^r) is a divisor of A002275(2^r*s), 10^(2^w) + 1 is also a divisor of A002275(2^r*s). For 1 <= m < 2^r, let t be the 2-adic valuation of m, i.e. 0 <= t = A007814(m) < r. Then 10^(2^r*s+m)+1 = 10^(2^t*q)+1 = (10^(2^t))^q + 1 for some odd number q. Since the sum of two odd powers a^q+b^q is divisible by a+b, this implies that 10^(2^r*s+m)+1 is divisible by 10^(2^t)+1. This means that for n = 2^r*s and 1 <= m < 2^r, gcd(A002275(n),10^(n+m)+1) >= 10^(2^t)+1 > 1, i.e. k//A002275(n)//k is not prime. Thus a(2^r*s) must have at least 2^r digits, i.e. a(2^r*s) >= 10^(2^r-1). QED As a consequence, a(n) >= 10^(A006519(n)-1). This result is still true if some of the digits of k are leading zeros. (End) LINKS Chai Wah Wu, Table of n, a(n) for n = 0..1023 FORMULA a(A004023(n)-2) = 1. - Chai Wah Wu, Nov 04 2019 EXAMPLE R_0 = 0 and the smallest k such that k//0//k is prime is 1, so a(0) = 1. MATHEMATICA Table[k = 1; While[! PrimeQ[f[n, k]], k++]; k, {n, 0, 7}] (* Michael De Vlieger, Oct 13 2015 *) PROG (PARI) a(n) = my(rep=(10^n-1)/9, k=1); while(!ispseudoprime(eval(Str(k, rep, k))), k++); k CROSSREFS Cf. A002275, A004023, A006519, A056251. Sequence in context: A006486 A273946 A070518 * A045525 A030774 A024685 Adjacent sequences: A263179 A263180 A263181 * A263183 A263184 A263185 KEYWORD nonn,base AUTHOR Felix Fröhlich, Oct 11 2015 EXTENSIONS a(16)-a(46) from Chai Wah Wu, Nov 04 2019 STATUS approved

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Last modified June 6 22:19 EDT 2023. Contains 363151 sequences. (Running on oeis4.)