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3/10: 3/40: 9/40 3/5: 3/10: −9/10: 6/5 1: −11/54: 5/2: −70/27: 35/27 7/8: 1631/55296: 175/512: 575/13824: 44275/110592: 253/4096: 37/378: 0: 250/621: 125/594: 0 ...
0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add" : a (0) = 0; for n > 0, a ( n ) = a ( n − 1) − n if that number is positive and not already in the sequence, otherwise a ( n ) = a ( n − 1) + n , whether or not that number is already in the sequence.
Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS). To illustrate: it can be seen that 16 is a superperfect number as σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16. If n is an even superperfect number, then n must be a power of 2, 2 k, such that 2 k+1 − 1 is a ...
Microsoft Math Solver (formerly Microsoft Mathematics and Microsoft Math) is an entry-level educational app that solves math and science problems. Developed and maintained by Microsoft , it is primarily targeted at students as a learning tool.
4095 – triangular number [2] and odd abundant number; [9] number of divisors in the sum of the fifth and largest known unitary perfect number, largest Ramanujan–Nagell number of the form [10] 4096 = 64 2 = 16 3 = 8 4 = 4 6 = 2 12 , smallest number with exactly 13 factors, a superperfect number [ 11 ]
Lotus, which had acquired Software Arts, including TK Solver, in 1984 [3] sold its ownership of the software to Universal Technical Systems less than two years later. [2] Release 5 was still considered "one of the longest–standing mathematical equation solvers on the market today" in 2012. [5] [6]
Z3 was open sourced in the beginning of 2015. [3] The source code is licensed under MIT License and hosted on GitHub. [4] The solver can be built using Visual Studio, a makefile or using CMake and runs on Windows, FreeBSD, Linux, and macOS. The default input format for Z3 is SMTLIB2.