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64
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Natural number
64 (sixty-four) is the natural number following 63 and preceding 65.
Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the seventeenth interprime, since it lies midway between the eighteenth and nineteenth prime numbers (61, 67).[1]
The aliquot sum of a power of two (2n) is always one less than the power of two itself, therefore the aliquot sum of 64 is 63, within an aliquot sequence of two composite members (64, 63, 41, 1, 0) that are rooted in the aliquot tree of the thirteenth prime, 41.[2]
64 is:
- the smallest number with exactly seven divisors,[3]
- the first whole number (greater than one) that is both a perfect square, and a perfect cube,[4]
- the lowest positive power of two that is not adjacent to either a Mersenne prime or a Fermat prime,
- the fourth superperfect number — a number such that σ(σ(n)) = 2n,[5]
- the sum of Euler's totient function for the first fourteen integers,[6]
- the number of graphs on four labeled nodes,[7]
- the index of Graham's number in the rapidly growing sequence 3↑↑↑↑3, 3 ↑3↑↑↑↑3 3, …
- the number of vertices in a 6-cube,
- the fourth dodecagonal number,[8]
- and the seventh centered triangular number.[9]
Since it is possible to find sequences of 65 consecutive integers (intervals of length 64) such that each inner member shares a factor with either the first or the last member, 64 is the seventh Erdős–Woods number.[10]
In decimal, no integer added to the sum of its own digits yields 64; hence, 64 is the tenth self number.[11]
In four dimensions, there are 64 uniform polychora aside from two infinite families of duoprisms and antiprismatic prisms, and 64 Bravais lattices.[12]
A chessboard has 64 squares.
- ^ Sloane, N. J. A. (ed.). "Sequence A024675 (Average of two consecutive odd primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.
- ^ Sloane, N. J. A., ed. (1975). "Aliquot sequences". The On-Line Encyclopedia of Integer Sequences. 29 (129). The OEIS Foundation: 101–107. Retrieved 2023-11-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A030516 (Numbers with 7 divisors. 6th powers of primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A019279 (Superperfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function: a(n) is Sum_{k equal to 1..n} phi(k), cf. A000010.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A006125 (a(n) equal to 2^(n*(n-1)/2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-16.
- ^ Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ "Sloane's A059756 : Erdős-Woods numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
- ^ "Sloane's A003052 : Self numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
- ^ Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans (1978), Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], ISBN 978-0-471-03095-9, MR 0484179
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