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Thread: Weird Sequence

  1. #1
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    Weird Sequence

    Let $\displaystyle a_1=1$. For all $\displaystyle n>1$, define

    $\displaystyle a_n=\sum_{\substack{i<n \\ \gcd(i,n)=1}}a_i$

    So $\displaystyle a_2=a_1$ and $\displaystyle a_4=a_1+a_3$.
    Find $\displaystyle a_{24}$
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  2. #2
    Moo
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    Hello,

    Let n be 24

    You're looking for $\displaystyle a_{24} = \sum_{\substack{i<24 \\ \gcd(i,24)=1}} a_i$

    All i<24 with gcd(i,24)=1 are :
    1,2,5,7,11,13,17,19,23

    Hence $\displaystyle a_{24} = a_1+a_2+a_5+a_7+a_{11}+a_{13}+a_{17}+a_{19}+a_{23}$

    We already know $\displaystyle a_1$ and $\displaystyle a_2$

    - let's calculate $\displaystyle a_5$
    $\displaystyle a_5 = \sum_{\substack{i<5 \\ \gcd(i,5)=1}}$
    All i<5 with gcd(i,5)=1 are : 1,2,3,4

    ----> $\displaystyle a_5 = a_1+a_2+a_3+a_4$

    -------> $\displaystyle a_{24} = a_1+a_2+a_1+a_2+a_3+a_4+a_7+a_{11}+a_{13}+a_{17}+a _{19}+a_{23}$

    $\displaystyle a_4=a_1+a_3$

    $\displaystyle a_{24} = 2 a_1+2a_2 + a_3+a_4+a_7+a_{11}+a_{13}+a_{17}+a_{19}+a_{23}$ $\displaystyle = 3 a_1+2a_2 + 2a_3+a_7+a_{11}+a_{13}+a_{17}+a_{19}+a_{23}$


    - let's calculate $\displaystyle a_3$
    All i<3 with gcd(i,3)=1 are : 1,2
    $\displaystyle a_3 = a_1+a_2$

    -------> $\displaystyle a_{24} = \dots$


    And so on... It's always the same thing : when there is an $\displaystyle a_i$ you don't know the value, get the definition, find all the numbers < i and relatively prime with i and continue...

    It's kinda hard to type it on a computer... If you really need all the details, i can help you for some following values
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  3. #3
    Member Henderson's Avatar
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    Maybe this will help you get started:

    1 1
    2 1
    3 2
    4 3
    5 7
    6 8
    7 22
    8 32
    9 66
    10 91
    11 233
    12 263
    13 729
    14 1038
    15 2059
    16 3119
    17 7674
    18 8666
    19 24014
    20 32741


    I can't promise this is correct; one small mistake would snowball into larger mistakes later.
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