# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a108582 Showing 1-1 of 1 %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, Table of n, a(n) for n = 1..8281 %H A108582 Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, 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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE