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A090739 Exponent of 2 in 9^n - 1.

%I A090739 #48 Jun 28 2025 10:24:34
%S A090739 3,4,3,5,3,4,3,6,3,4,3,5,3,4,3,7,3,4,3,5,3,4,3,6,3,4,3,5,3,4,3,8,3,4,
%T A090739 3,5,3,4,3,6,3,4,3,5,3,4,3,7,3,4,3,5,3,4,3,6,3,4,3,5,3,4,3,9,3,4,3,5,
%U A090739 3,4,3,6,3,4,3,5,3,4,3,7,3,4,3,5,3,4,3
%N A090739 Exponent of 2 in 9^n - 1.
%C A090739 The exponent of 2 in the factorization of Fibonacci(6n). - _T. D. Noe_, Mar 14 2014
%C A090739 Records of 3, 4, 5, 6, 7, 8,.. occur at n= 1, 2, 4, 8, 16, 32,... - _R. J. Mathar_, Jun 28 2025
%H A090739 T. D. Noe, <a href="/A090739/b090739.txt">Table of n, a(n) for n = 1..1000</a>
%H A090739 T. Lengyel, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-3/lengyel.pdf">The order of the Fibonacci and Lucas numbers</a>, Fib. Quart. 33 (1995), 234-239.
%F A090739 a(n) = A007814(n) + 3.
%F A090739 a((2*n-1)*2^p) = p + 3, p >= 0. - _Johannes W. Meijer_, Feb 08 2013
%F A090739 a(n) = log_2(A006519(9^n - 1)). - _Alonso del Arte_, Feb 08 2013
%F A090739 a(n) = 2*tau(4*n)/(tau(4*n) - tau(n)), where tau(n) = A000005(n). - _Peter Bala_, Jan 06 2021
%F A090739 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - _Amiram Eldar_, Nov 28 2022
%e A090739 For n = 2, we see that -1 + 3^4 = 80 = 2^4 * 5 so a(2) = 4.
%e A090739 For n = 3, we see that -1 + 3^6 = 728 = 2^3 * 7 * 13, so a(3) = 3.
%p A090739 A090739 := proc(n)
%p A090739     padic[ordp](9^n-1,2) ;
%p A090739 end proc:
%p A090739 seq(A090739(n),n=1..80) ; # _R. J. Mathar_, Jun 28 2025
%t A090739 Table[Part[Flatten[FactorInteger[ -1+3^(2*n)]], 2], {n, 1, 70}]
%t A090739 Table[IntegerExponent[Fibonacci[n], 2], {n, 6, 600, 6}] (* _T. D. Noe_, Mar 14 2014 *)
%o A090739 (PARI) a(n)=valuation(n,2)+3 \\ _Charles R Greathouse IV_, Mar 14 2014
%o A090739 (Python)
%o A090739 def A090739(n): return (~n&n-1).bit_length()+3 # _Chai Wah Wu_, Jul 11 2022
%Y A090739 Cf. A024101, A069895, A091512, A088660, A090740, A220466.
%Y A090739 Cf. A000005, A006519, A120738 (partial sums).
%Y A090739 Appears in A161737.
%K A090739 nonn,easy
%O A090739 1,1
%A A090739 _Labos Elemer_ and _Ralf Stephan_, Jan 19 2004
%E A090739 More terms from _T. D. Noe_, Mar 14 2014