64 (number)

Mathematics

64 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).

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.

64 is:

• the smallest number with exactly seven divisors,
• the first whole number (greater than one) that is both a perfect square, and a perfect cube,
• 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,
• the sum of Euler's totient function for the first fourteen integers,
• the number of graphs on four labeled nodes,
• the index of Graham's number in the rapidly growing sequence 3↑↑↑↑3, 3 ↑ 3, ...
• the number of vertices in a 6-cube,
• the fourth dodecagonal number,
• and the seventh centered triangular number.

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.

In decimal, no integer added to the sum of its own digits yields 64; hence, 64 is the tenth self number.

In four dimensions, there are 64 uniform polychora aside from two infinite families of duoprisms and antiprismatic prisms, and 64 Bravais lattices.

References

References

1. {{Cite OEIS. A024675. Average of two consecutive odd primes.
2. (1975). "Aliquot sequences". The [[OEIS Foundation]].
3. {{Cite OEIS. A005179. Smallest number with exactly n divisors
4. {{Cite OEIS. A030516. Numbers with 7 divisors. 6th powers of primes
5. {{Cite OEIS. A019279. Superperfect numbers
6. {{Cite OEIS. A002088. Sum of totient function: a(n) is Sum_{k equal to 1..n} phi(k), cf. A000010.
7. {{Cite OEIS. A006125. a(n) equal to 2^(n*(n-1)/2).
8. {{Cite OEIS. A051624. 12-gonal (or dodecagonal) numbers
9. {{Cite OEIS. A005448. Centered triangular numbers
10. "Sloane's A059756 : Erdős-Woods numbers". OEIS Foundation.
11. "Sloane's A003052 : Self numbers". OEIS Foundation.
12. (1978). "Crystallographic groups of four-dimensional space". Wiley-Interscience [John Wiley & Sons].