Thabit number
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A Thabit number is an integer of the form [3 \cdot 2^n - 1]. The first few Thabit numbers are:
2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863
The binary representation of Thabit numbers is n + 2 digits long, consisting of "10" followed by n 1s. Thus, with the exception of 5, no Thabit number is palindromic in binary.
When both n and n - 1 yield prime Thabit numbers, and [9 \cdot 2^ - 1] is also prime, a pair of amicable numbers can be calculated, the abundant member of the pair given by the formula [2^n(3 \cdot 2^ - 1)(3 \cdot 2^n - 1)] and the deficient member of the pair is given by the formula [2^n(9 \cdot 2^ - 1)].
So, for example, n = 2 gives the Thabit number 11, and n = 1 gives the Thabit number 5, and our third term is 71. Then, 22 = 4, multiplied by 5 and 11 results in 220, whose divisors add up to 284, and 4 times 71 is 284, whose divisors add up to 220.
The only n meeting these conditions are 2, 4 and 7, corresponding to the Thabit numbers 11, 47 and 383.
The 9th Century astronomer Thabit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers.
As of December 2005, the largest known Thabit prime corresponds to n = 2312734, a number of almost 700000 digits in base 10.
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