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.

The first few Thabit numbers that are prime (sequence A007505 in OEIS):

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831

As of November 2006, the known n values which give prime Thabit numbers are: [1]

n = 0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734

The primes from n = 234760 were found by the distributed computing project "321 search". [2] The largest of these, 3 · 2^2312734 - 1, has 696203 digits and was found by Paul Underwood in 2005.

When both n and n - 1 yield prime Thabit numbers, and $9 \cdot 2^{2n - 1} - 1$ is also prime, a pair of amicable numbers can be calculated, the abundant member of the pair given by the formula $2^n(3 \cdot 2^{n - 1} - 1)(3 \cdot 2^n - 1)$ and the deficient member of the pair is given by the formula $2^n(9 \cdot 2^{2n - 1} - 1)$ .

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.