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Given a sequence S_i as input to stage i, form sequence S_(i+1) as follows: 1. For k in [1,...,i], write term i+k and then term i-k. 2. Discard the ith term. 3. Write the ...

The problem of determining how many nonattacking kings can be placed on an n×n chessboard. For n=8, the solution is 16, as illustrated above (Madachy 1979). In general, the ...

The thinnest sequence which contains 1, and whenever it contains x, also contains 2x, 3x+2, and 6x+3: 1, 2, 4, 5, 8, 9, 10, 14, 15, 16, 17, ... (OEIS A005658).

Let phi_x^((k)) denote the recursive function of k variables with Gödel number x, where (1) is normally omitted. Then if g is a partial recursive function, there exists an ...

For every k>=1, let C_k be the set of composite numbers n>k such that if 1<a<n, GCD(a,n)=1 (where GCD is the greatest common divisor), then a^(n-k)=1 (mod n). Special cases ...

The numbers defined by the recurrence relation K_(n+1)=1+min(2K_(|_n/2_|),3K_(|_n/3_|)), with K_0=1. The first few values for n=0, 1, 2, ... are 1, 3, 3, 4, 7, 7, 7, 9, 9, ...

Knuth's series is given by S = sum_(k=1)^(infty)((k^k)/(k!e^k)-1/(sqrt(2pik))) (1) = -2/3-1/(sqrt(2pi))zeta(1/2) (2) = -0.08406950872765599646... (3) (OEIS A096616), where ...

The first few numbers whose abundance absolute values are odd squares (excluding the trivial cases of powers of 2) are 98, 2116, 4232, 49928, 80656, 140450, 550564, 729632, ...

Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the number of nonisomorphic Abelian finite groups ...

If theta is a given irrational number, then the sequence of numbers {ntheta}, where {x}=x-|_x_|, is dense in the unit interval. Explicitly, given any alpha, 0<=alpha<=1, and ...