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%I #74 Sep 08 2022 08:44:40
%S 0,0,2,0,2,4,2,0,8,4,2,4,2,4,8,0,2,10,2,16,8,4,2,16,7,4,26,16,2,4,2,0,
%T 8,4,18,28,2,4,8,16,2,22,2,16,17,4,2,16,30,24,8,16,2,28,43,32,8,4,2,
%U 16,2,4,8,0,32,64,2,16,8,44,2,64,2,4,68,16,18,64,2,16,80,4,2,64
%N a(n) = 2^n mod n.
%C 2^n == 2 mod n if and only if n is a prime or a member of A001567 or of A006935. [Guy]. - _N. J. A. Sloane_, Mar 22 2012; corrected by _Thomas Ordowski_, Mar 26 2016
%C Known solutions to 2^n == 3 (mod n) are given in A050259.
%C This sequence is conjectured to include every integer k >= 0 except k = 1. A036236 includes a proof that k = 1 is not in this sequence, and n = A036236(k) solves a(n) = k for all other 0 <= k <= 1000. - _David W. Wilson_, Oct 11 2011
%C It could be argued that a(0) := 1 would make sense, e.g., thinking of "mod n" as "in Z/nZ", and/or because (anything)^0 = 1. See also A112987. - _M. F. Hasler_, Nov 09 2018
%D Richard K. Guy, Unsolved Problems in Number Theory, F10.
%H T. D. Noe, <a href="/A015910/b015910.txt">Table of n, a(n) for n=1..10000</a>
%H Albert Frank, <a href="http://www.paulcooijmans.com/oth/intcont2003.html">International Contest Of Logical Sequences</a>, 2002 - 2003. Item 4
%H Albert Frank, <a href="http://www.paulcooijmans.com/oth/intcont2003ans.html">Solutions of International Contest Of Logical Sequences</a>, 2002 - 2003.
%H Peter L. Montgomery, <a href="http://www.cs.ucla.edu/~klinger/newno.html">65-digit solution</a>.
%F a(2^k) = 0. - _Alonso del Arte_, Nov 10 2014
%F a(n) == 2^(n-phi(n)) mod n, where phi(n) = A000010(n). - _Thomas Ordowski_, Mar 26 2016
%e a(7) = 2 because 2^7 = 128 = 2 mod 7.
%e a(8) = 0 because 2^8 = 256 = 0 mod 8.
%e a(9) = 8 because 2^9 = 512 = 8 mod 9.
%p A015910 := n-> modp(2 &^ n,n) ; # _Zerinvary Lajos_, Feb 15 2008
%t Table[PowerMod[2, n, n], {n, 85}]
%o (Maxima) makelist(power_mod(2, n, n), n, 1, 84); /* _Bruno Berselli_, May 20 2011 */
%o (PARI) a(n)=lift(Mod(2,n)^n) \\ _Charles R Greathouse IV_, Jul 15 2011
%o (Haskell)
%o import Math.NumberTheory.Moduli (powerMod)
%o a015910 n = powerMod 2 n n -- _Reinhard Zumkeller_, Oct 17 2015
%o (Magma) [Modexp(2, n, n): n in [1..100]]; // _Vincenzo Librandi_, Nov 09 2018
%Y Cf. A036236, A015911, A015921, A001567.
%Y Cf. A000079, A062173, A082495.
%K nonn
%O 1,3
%A _Robert G. Wilson v_
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