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%I #15 Jul 20 2021 03:40:45
%S 2,2,4,64,4096,1152921504606846976,1152921504606846976
%N 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.
%C 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.
%C a(7) = 2^420 contains 127 decimal digits.
%C From _Jianing Song_, Jul 20 2021: (Start)
%C 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.
%C 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.
%C Obviously, here 2 can be replaced by any prime p provided that {a(n)} is defined as a(n) = p^A003418(n). (End)
%H Jianing Song, <a href="/A178981/b178981.txt">Table of n, a(n) for n = 0..10</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%e 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.
%o (PARI) a(n)=2^(lcm(vector(n, i, i))) \\ _Jianing Song_, Jul 20 2021, following a PARI program for A003418
%Y Cf. A003418, A050923.
%K nonn,easy
%O 0,1
%A _Rick L. Shepherd_, Jan 02 2011
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