A178981 2^A003418(n); for n >= 1, the least number > 1 that can be expressed simultaneously as a k-th power of some integer for all 1 <= k <= n.

2, 2, 4, 64, 4096, 1152921504606846976, 1152921504606846976

Offset: 0

Author

Rick L. Shepherd, Jan 02 2011

Keywords

• nonn
• easy

Comments

Equivalently, for n >= 1, the least number > 1 of objects that can be arranged as a k-cube (k-dimensional hypercube) for all 1 <= k <= n.

a(7) = 2^420 contains 127 decimal digits.

From Jianing Song, Jul 20 2021: (Start)

Let F_q be the finite field with q elements, then F_a(n) is the smallest extension field of F_2 such that every polynomial of degree at most n splits into linear factors.

Union_{n>=0} F_a(n) is the algebraic clousre of F_2, which is the unique algebraically closed field with characteristic 2 and transcendence degree 0 (note that an algebraically closed field is uniquely determined by its characteristic and transcendence degree). Union_{n>=0} F_(2^(n!)) = Union_{n>=0} F_A050923(n) gives the same field.

Obviously, here 2 can be replaced by any prime p provided that {a(n)} is defined as a(n) = p^A003418(n). (End)

Examples

			a(6) = 2^A003418(6) = 2^60 = 1152921504606846976 [= (2^60)^1] = (2^30)^2 = 1073741824^2 = (2^20)^3 = 1048576^3 = (2^15)^4 = 32768^4 = (2^12)^5 = 4096^5 = (2^10)^6 = 1024^6, while no smaller integer > 1 can be expressed simultaneously as a square, cube, 4th, 5th, and 6th power of integers.
		

Links

• Jianing Song, Table of n, a(n) for n = 0..10
• Index to divisibility sequences

Crossrefs

Cf. A003418, A050923.

Programs

PARI

a(n)=2^(lcm(vector(n, i, i))) \\ Jianing Song, Jul 20 2021, following a PARI program for A003418