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  2. 1 + 2 + 4 + 8 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E...

    In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity. However, it can be manipulated to yield a number of ...

  3. List of integer sequences - Wikipedia

    en.wikipedia.org/wiki/List_of_integer_sequences

    Name First elements Short description OEIS Mersenne prime exponents : 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... Primes p such that 2 p − 1 is prime.: A000043 ...

  4. Power of two - Wikipedia

    en.wikipedia.org/wiki/Power_of_two

    1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (sequence A000079 in the OEIS) By comparison, powers of two with negative exponents are fractions: for a negative integer n, 2 n is one half multiplied by itself n times. Thus the first few powers of two where n is negative are ⁠ 1 / 2 ⁠, ⁠ 1 / 4 ⁠, ⁠ 1 / 8 ⁠, ⁠ 1 / 16 ⁠, etc.

  5. 1/2 + 1/4 + 1/8 + 1/16 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/...

    1/2 + 1/4 + 1/8 + 1/16 + ⋯. First six summands drawn as portions of a square. The geometric series on the real line. In mathematics, the infinite series ⁠ 1 2 ⁠ + ⁠ 1 4 ⁠ + ⁠ 1 8 ⁠ + ⁠ 1 16 ⁠ + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation ...

  6. Wheat and chessboard problem - Wikipedia

    en.wikipedia.org/wiki/Wheat_and_chessboard_problem

    The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8 + ... and so forth for the 64 squares. The total number of grains can be shown to be 2 641 or 18,446,744,073,709,551,615 (eighteen quintillion ...

  7. Fermat number - Wikipedia

    en.wikipedia.org/wiki/Fermat_number

    In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form: where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS ). If 2 k + 1 is prime and k > 0, then k itself ...

  8. 1 − 2 + 4 − 8 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_4_%E2%88...

    12 + 48 + ⋯. In mathematics, 12 + 48 + ⋯ is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. As a series of real numbers, it diverges. So in the usual sense it has no sum.

  9. Complete sequence - Wikipedia

    en.wikipedia.org/wiki/Complete_sequence

    In mathematics, a sequence of natural numbers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. For example, the sequence of powers of two (1, 2, 4, 8, ...), the basis of the binary numeral system, is a complete sequence; given any natural number, we can ...