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A045616
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Primes p such that 10^(p-1) == 1 (mod p^2).
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19
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OFFSET
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1,1
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COMMENTS
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Primes p such that the decimal fraction 1/p has same period length as 1/p^2, i.e., the multiplicative order of 10 modulo p is the same as the multiplicative order of 10 modulo p^2. [extended by Felix Fröhlich, Feb 05 2017]
No further terms below 1.172*10^14 (as of Feb 2020, cf. Fischer's table).
56598313 was announced in the paper by Brillhart et al. - Helmut Richter, May 17 2004
A265012(A049084(a(n))) = 1. - Reinhard Zumkeller, Nov 30 2015
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REFERENCES
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J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, pp. 213-222 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, A3.
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LINKS
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Table of n, a(n) for n=1..3.
Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2).
Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
Peter L. Montgomery, New solutions of a^(p-1) == 1 (mod p^2), Math. Comp. 61 (1993), 361-363.
Math Overflow, Is the smallest primitive root modulo p a primitive root modulo p^2?, Jun 09 2010.
Helmut Richter, The period length of the decimal expansion of a fraction.
Helmut Richter, The Prime Factors Of 10^486-1.
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.
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MATHEMATICA
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A045616Q = PrimeQ@# && PowerMod[10, # - 1, #^2] == 1 &; Select[Range[1000000], A045616Q] (* JungHwan Min, Feb 04 2017 *)
Select[Prime[Range[34*10^5]], PowerMod[10, #-1, #^2]==1&] (* Harvey P. Dale, Apr 10 2018 *)
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PROG
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(PARI) lista(nn) = forprime(p=2, nn, if (Mod(10, p^2)^(p-1)==1, print1(p, ", "))); \\ Michel Marcus, Aug 16 2015
(Haskell)
import Math.NumberTheory.Moduli (powerMod)
a045616 n = a045616_list !! (n-1)
a045616_list = filter
(\p -> powerMod 10 (p - 1) (p ^ 2) == 1) a000040_list'
-- Reinhard Zumkeller, Nov 30 2015
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CROSSREFS
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Cf. A001220, A014127, A123692, A212583, A123693, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A039951.
Cf. A265012, A049084, A000040.
Sequence in context: A230029 A238447 A241977 * A198705 A198624 A198652
Adjacent sequences: A045613 A045614 A045615 * A045617 A045618 A045619
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KEYWORD
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bref,hard,nonn,nice,more
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AUTHOR
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Helmut Richter, Dec 11 1999
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STATUS
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approved
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