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n appears n^3 times.
2

%I #41 Nov 05 2024 12:17:51

%S 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 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 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 n appears n^3 times.

%C From _Jonathan Vos Post_, Mar 18 2006: (Start)

%C The key to this sequence is: 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+3+...+n)^2.

%C 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 The current sequence is, loosely, the inverse function of the square of the triangular number sequence. (End)

%H Boris Putievskiy, <a href="/A108582/b108582.txt">Table of n, a(n) for n = 1..8281</a>

%H 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 a(n) = ceiling((1/2)*(sqrt(8*sqrt(n) + 1) - 1)). - _Boris Putievskiy_, Jun 19 2024

%F From _Chai Wah Wu_, Nov 04 2024: (Start)

%F a(n) = m+1 if n>(m(m+1))^2/4 and a(n) = m otherwise where m = floor((4n)^(1/4)).

%F 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 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 Flatten @ Table[ Table[k, {k^3}], {k, 5}] (* _Giovanni Resta_, Jun 17 2016 *)

%t a[n_]:=Ceiling[1/2 (Sqrt[8 Sqrt[n]+1]-1)]

%t Nmax=225; Table[a[n],{n,1,Nmax}] (* _Boris Putievskiy_, Jun 19 2024 *)

%o (Python)

%o from sympy import integer_nthroot

%o def A108582(n): return (m:=integer_nthroot(k:=n<<2,4)[0])+(k>(m*(m+1))**2) # _Chai Wah Wu_, Nov 04 2024

%Y Cf. A000027, A000578, A002024, A072649, A074279, A000217.

%Y Cf. A000330, A000537, A006331, A050446, A050447, A006003, A005900.

%K easy,nonn,changed

%O 1,2

%A _Jonathan Vos Post_, Jul 25 2005

%E Two missing terms from _Giovanni Resta_, Jun 17 2016