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 A090129 Smallest exponent such that -1 + 3^a(n) is divisible by 2^n. 18
 1, 2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A131577 and A011782 are companions, A131577(n) + A011782(n) = 2^n, (and differences each other). - Paul Curtz, Jan 18 2009 A090127 with offset 0: (1, 2, 2, 4, 8, ...) = A(x) / A(x^2), when A(x) = (1 + 2x + 4x^2 + 8x^3 + ...). - Gary W. Adamson, Feb 20 2010 From Wolfdieter Lang, Apr 18 2012: (Start) a(n) is the order of 3 modulo 2^n. For n=1 and 2 this is obviously 1 and 2, respectively, and for n >= 3 it is 2^(n-2). For a proof see, e.g., the Graeme McRae link under A068531, the section 'A Different Approach', proposed by Alexander Monnas, the first part, where the result from the expansion of (4-1)^(2^(k-2)) holds only for k >= 3. See also the Charles R Greathouse IV program below where this result has been used. This means that the cycle generated by 3, taken modulo 2^n, has length a(n), and that 3 is not a primitive root modulo 2^n, if n >= 3 (because Euler's phi(2^n) = 2^(n-1), n >= 1, see A000010). (End) Let r(x) = (1 + 2x + 2x^2 + 4x^3 + ...). Then (1 + 2x + 4x^2 + 8x^3 + ...) = (r(x) * r(x^2) * r(x)^4 * r(x^8) * ...). - Gary W. Adamson, Sep 13 2016 LINKS Table of n, a(n) for n=1..36. Index to divisibility sequences FORMULA a(n) = 2^(n-2) if n >= 3, 1 for n=1 and 2 for n=2 (see the order comment above). a(n+2) = A152046(n) + A152046(n+1) = 2*A011782(n). - Paul Curtz, Jan 18 2009 EXAMPLE a(1) = 1 since -1 + 3 = 2 is divisible by 2^1; a(2) = a(3) = 2 since -1 + 9 = 8 is divisible by 4 = 2^2 and also by 8 = 2^3; a(5) = 8 since -1 + 6561 = 6560 = 32*205 is divisible by 2^5. From Wolfdieter Lang, Apr 18 2012: (Start) n=3: the order of 3 (mod 8) is a(3)=2 because the cycle generated by 3 is [3, 3^2==1 (mod 8)]. n=5: a(5) = 2^3 = 8 because the cycle generated by 3 is [3^1=3, 3^2=9, 3^3=27, 17, 19, 25, 11, 1] (mod 32). The multiplicative group mod 32 is non-cyclic (see A033949(10)) with the additional four cycles [5, 25, 29, 17, 21, 9, 13, 1], [7, 17, 23, 1], [15, 1], and [31, 1]. This is the cycle structure of the (Abelian) group Z_8 x Z_2 (see one of the cycle graphs shown in the Wikipedia link 'List of small groups' for the order phi(32)=16, given under A192005). (End) MATHEMATICA t=Table[Part[Flatten[FactorInteger[ -1+3^(n)]], 2], {n, 1, 130}] Table[Min[Flatten[Position[t, j]]], {j, 1, 10}] Join[{1, 2}, 2^Range[30]] (* or *) Join[{1, 2}, NestList[2#&, 2, 30]] (* Harvey P. Dale, Nov 08 2012 *) PROG (PARI) a(n)=2^(n+(n<3)-2) \\ Charles R Greathouse IV, Apr 09 2012 (Python) def A090129(n): return n if n<3 else 1<

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Last modified August 14 12:42 EDT 2024. Contains 375164 sequences. (Running on oeis4.)