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Number of powers of 9 modulo n.
16

%I #16 Aug 25 2024 10:00:50

%S 1,1,2,1,2,2,3,1,2,2,5,2,3,3,3,2,8,2,9,2,4,5,11,2,10,3,3,3,14,3,15,4,

%T 6,8,6,2,9,9,4,2,4,4,21,5,3,11,23,3,21,10,9,3,26,3,10,3,10,14,29,3,5,

%U 15,4,8,6,6,11,8,12,6,35,2,6,9,11,9,15,4,39,2,3,4,41,4,8,21,15,5,44,3,3

%N Number of powers of 9 modulo n.

%H Amiram Eldar, <a href="/A054717/b054717.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from David W. Wilson)

%F a(n) = valuation(3*n, 9) + A007740(n). - _Amiram Eldar_, Aug 25 2024

%e Take the sequence 1, 9, 81, 729, ... and reduce mod n; count distinct terms. For n = 5 we get 1, 4, 1, 4, ... so a(5) = 2.

%t With[{p9=9^Range[0,50]},Table[Length[Union[Mod[#,n]&/@p9]],{n,100}]] (* _Harvey P. Dale_, Apr 22 2012 *)

%t a[n_] := IntegerExponent[3*n, 9] + MultiplicativeOrder[9, n/3^IntegerExponent[n, 3]]; Array[a, 100] (* _Amiram Eldar_, Aug 25 2024 *)

%Y Cf. A007740.

%Y Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

%K easy,nonn

%O 1,3

%A _Henry Bottomley_, Apr 20 2000