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 Missing Numbers Please (Posted on 2007-08-21)
What are the missing numbers in this sequence:
0, __, 2, __, 9, __, 20, __, 26, __, 54, __, 77, __

 See The Solution Submitted by Brian Smith Rating: 4.3333 (6 votes)

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 Puzzle Solution Comment 7 of 7 |

Let us denote the pth term of the given sequence as S(p).

Let us consider p = 3. The only numbers (d) 1< d < 3, with gcd(d, 3) = 1, are 1 and 2, and we observe that 1+2 = 3 is the third term.

For p =5, the only numbers d in the range 1 < d< 5, that are coprime to d are 1,2,3 and 4, the sum of which is 9, and we observe that 9 corresponds to the fifth term.

We can now conjecture that S(p) = Sum of the d's in the range 1< d < p, such that gcd(d, p) = 1

This conjecture is verified to be true for the other given values of p = 7, 9, 11, 13

For example, the valid values of d satisfying 1< d< 13, with gcd(d, 13) = 1 are d = 2,3,4,5,6,7,8,9,10,11,12. Summing these d values, we otain 17, which is the 13th term.

It now remains to determine S(p) for p = 2,4,6,8,10,12,14

As before, finding the possible d values in the appropriate range and summing over them, we obtain the required missing terms as follows:

S(2) = 0
S(4) = 2+3 = 5
S(6) = 5
S(8) = 3+5+7 = 15
S(10) = 3+7+9 = 19
S(12) = 5+7+11 = 23
S(14) = 3+5+7+9+11+13 = 41

Consequently, The required missing terms in this order are
0, 3, 5, 15, 19, 23, 41

Edited on July 6, 2008, 4:24 pm
 Posted by K Sengupta on 2008-06-15 00:40:20

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