This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A108582 #41 Nov 05 2024 12:17:51
%S A108582 1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%T A108582 3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A108582 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5
%N A108582 n appears n^3 times.
%C A108582 From _Jonathan Vos Post_, Mar 18 2006: (Start)
%C A108582 The key to this sequence is: 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+3+...+n)^2.
%C A108582 Since the last occurrence of n comes one before the first occurrence of n+1 and the former is at Sum_{i=0..n} i^3 = A000537(n) = (A000217(n))^2 = (n*(n+1)/2)^2 = (C(n+1,2))^2, have a(A000537(n)) = a((A000217(n))^2) = n and thus a(1+A000537(n)) = a(1+(A000217(n))^2) = n+1.
%C A108582 The current sequence is, loosely, the inverse function of the square of the triangular number sequence. (End)
%H A108582 Boris Putievskiy, <a href="/A108582/b108582.txt">Table of n, a(n) for n = 1..8281</a>
%H A108582 Boris Putievskiy, <a href="https://arxiv.org/abs/2310.18466">Integer Sequences: Irregular Arrays and Intra-Block Permutations</a>, arXiv:2310.18466 [math.CO], 2023.
%F A108582 a(n) = ceiling((1/2)*(sqrt(8*sqrt(n) + 1) - 1)). - _Boris Putievskiy_, Jun 19 2024
%F A108582 From _Chai Wah Wu_, Nov 04 2024: (Start)
%F A108582 a(n) = m+1 if n>(m(m+1))^2/4 and a(n) = m otherwise where m = floor((4n)^(1/4)).
%F A108582 More generally, for a sequence a_k(n) where n appears n^(k-1) times, a_k(n) = m+1 if n > Sum_{i=1..m} i^(k-1) and a_k(n) = m otherwise where m = floor((kn)^(1/k)).
%F A108582 Note that Sum_{i=1..m} i^(k-1) can be written as a k-th order polynomial of m using Faulhaber's formula. (End)
%t A108582 Flatten @ Table[ Table[k, {k^3}], {k, 5}] (* _Giovanni Resta_, Jun 17 2016 *)
%t A108582 a[n_]:=Ceiling[1/2 (Sqrt[8 Sqrt[n]+1]-1)]
%t A108582 Nmax=225; Table[a[n],{n,1,Nmax}] (* _Boris Putievskiy_, Jun 19 2024 *)
%o A108582 (Python)
%o A108582 from sympy import integer_nthroot
%o A108582 def A108582(n): return (m:=integer_nthroot(k:=n<<2,4)[0])+(k>(m*(m+1))**2) # _Chai Wah Wu_, Nov 04 2024
%Y A108582 Cf. A000027, A000578, A002024, A072649, A074279, A000217.
%Y A108582 Cf. A000330, A000537, A006331, A050446, A050447, A006003, A005900.
%K A108582 easy,nonn
%O A108582 1,2
%A A108582 _Jonathan Vos Post_, Jul 25 2005
%E A108582 Two missing terms from _Giovanni Resta_, Jun 17 2016