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Let S(x) denote the number of positive integers not exceeding x which can be expressed as a sum of two squares (i.e., those n<=x such that the sum of squares function ...

A zerofree number n is called right truncatable if n and all numbers obtained by successively removing the rightmost digits are prime. There are exactly 83 right truncatable ...

cos(pi/(16)) = 1/2sqrt(2+sqrt(2+sqrt(2))) (1) cos((3pi)/(16)) = 1/2sqrt(2+sqrt(2-sqrt(2))) (2) cos((5pi)/(16)) = 1/2sqrt(2-sqrt(2-sqrt(2))) (3) cos((7pi)/(16)) = ...

The complexity c_n of an integer n is the least number of 1s needed to represent it using only additions, multiplications, and parentheses. For example, the numbers 1 through ...

Given two starting numbers (a_1,a_2), the following table gives the unique sequences {a_i} that contain no three-term arithmetic progressions. Sloane sequence A003278 1, 2, ...

Ramanujan developed a number of interesting closed-form expressions for generalized continued fractions. These include the almost integers ...

Let the stick number s(K) of a knot K be the least number of straight sticks needed to make a knot K. The smallest stick number of any knot is s(T)=6, where T is the trefoil ...

A number which is simultaneously octagonal and pentagonal. Let O_n denote the nth octagonal number and P_m the mth pentagonal number, then a number which is both octagonal ...

A cubic number is a figurate number of the form n^3 with n a positive integer. The first few are 1, 8, 27, 64, 125, 216, 343, ... (OEIS A000578). The protagonist Christopher ...

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given by 668+866=1534, so the next ...