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A039951
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a(n) is the smallest prime p such that p^2 divides n^(p-1) - 1.
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40
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2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3
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OFFSET
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1,1
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COMMENTS
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a(n^k) <= a(n) for any n,k > 1.
a(n) is currently unknown for n in {47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, ...}. - Richard Fischer, Jul 15 2021
a(47) > 1.4*10^14, a(72) > 1.4*10^14 (see Fischer's tables).
For all nonnegative integers n and k, a(n^(n^k)) = a(n) (see Puzzle 762 in the links). Also a(n) = 3 if and only if mod(n, 36) is in the set {8, 10, 19, 26, 28, 35}. - Farideh Firoozbakht and Jahangeer Kholdi, Nov 01 2014
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LINKS
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FORMULA
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a(4k+1) = 2.
a(n) = A096082(n) for all n > 1 that are not of the form 4k+1. Note that A096082 begins with n = 2. [Corrected and clarified by Jonathan Sondow, Jun 17-18 2010]
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MATHEMATICA
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Table[p = 2; While[! Divisible[n^(p - 1) - 1, p^2], p = NextPrime@ p]; p, {n, 33}] (* Michael De Vlieger, Nov 24 2016 *)
f[n_] := Block[{p = 2}, While[ PowerMod[n, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 33] (* Robert G. Wilson v, Jul 18 2018 *)
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PROG
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(PARI) for(n=1, 20, forprime(p=2, 1e9, if(Mod(n, p^2)^(p-1)==1), print1(p, ", "); next({2}))); print1("--, ")) \\ Felix Fröhlich, Jul 24 2014
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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EXTENSIONS
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a(34)-a(46) from Helmut Richter (richter(AT)lrz.de), May 17 2004
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STATUS
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approved
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