Tag Archives: Sequence

Integers of the form (x^2 – 12)/y and (y^2 – 12)/x, x ≠ y

    Find two distinct positive integers     and     such that Each of     and     is an integer.     Here are the first few solutions:     For all members of the sequence, … Continue reading

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Sequence : S_{n+3} = 2(S_{n+2}) + 2(S_{n+1}) – S_{n}

    The sequence     of Fibonacci numbers is defined by the recurrence relation with seed values ,    or ,      Let   ,    ,   and     for         Is     always … Continue reading

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Integers n; sum of the reciprocals of prime factors + 1/n = 1

        Allowing prime powers among the divisors k = 7,    n = 144508961850 Prime factors :    (2,3,11,25,29,1097,2753) The sum of the reciprocals (S) is, 1/2 + 1/3 + 1/11 + 1/25 + 1/29 + 1/1097 … Continue reading

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Sequence 91, 8911, 889111, 88891111, …

    Show that the sequence   (putting one more 8 on the front and one more 1 on the end) consists solely of triangular numbers:                         ………………………….. …………………………..                   … Continue reading

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Sequences of squares: a+1, ab+1, abc+1, abcd+1, …

  There exist an infinite number of infinite sequences of distinct positive integers    , … for which     , … are all squares.     then,     then,     then,     then,   etc.   … Continue reading

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Table | integers 1, 2, 3,…, k^2

  Select an arbitrary number and delete from the table the row and the column which contain this number. Do the same with the remaining … table of   25   numbers, etc.,   6 times.            (Table #1) … … Continue reading

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A sequence of fractions

  Given any two successive terms   a,   b,   of the sequence, the next term is obtained as shown below. Explain what’s happening here.                    

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Sequence of prime numbers: p_(k+1) – (p_k + p_(k-1))= A

  Find a sequence of prime numbers ,   ,   ,   … ,   ,   ,   ,   … ,   such that and                         … Continue reading

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Sequence sin (3n)

             

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Sequence of (2^n – 1) consecutive integers all having n factors

  The Goal: To find a sequence of  (2^n – 1)  consecutive integers such that each of them has n prime factors   If   n = 2 The smallest sequence of  (2^2 – 1 = 3)  consecutive integers such that … Continue reading

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