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 A255165 #46 Sep 08 2022 08:46:11
%S A255165 0,1,2,4,5,6,7,9,11,12,13,14,15,16,17,20,21,22,23,24,25,26,27,28,30,
%T A255165 31,33,34,35,36,37,39,40,41,42,44,45,46,47,48,49,50,51,52,53,54,55,56,
%U A255165 58,59,60,61,62,63,64,65,66,67,68,69,70,71,72
%N A255165 a(n) = Sum_{k=2..n} floor(log(n)/log(k)), n >= 1.
%C A255165 The sum jumps up by 2 or more where n is a power of one or more k < n, otherwise it gains 1 with each increase in n.
%C A255165 First differences = A089723.
%C A255165 This calculation is analogous to that used for the sum of the number of divisors for all integers <= n in A006218.
%C A255165 a(n)+n gives the number of digits in the representations of n from base 2 to base n+1. - _Christina Steffan_, Dec 06 2015
%C A255165 Without floor, Sum_{k=2..n} log(n)/log(k) ~ n * (1 + 1/log(n) + 2/log(n)^2 + 6/log(n)^3 + 24/log(n)^4 + 120/log(n)^5 + ...). - _Vaclav Kotesovec_, Apr 06 2021
%H A255165 Vaclav Kotesovec, <a href="/A255165/b255165.txt">Table of n, a(n) for n = 1..10000</a>
%H A255165 Jan Mycielski, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm2/cm2140.pdf">Sur les représentations des nombres naturels par des puissances à base et exposant naturels</a>, Colloquium Mathematicum 2 (1951), 254-260. See T(n) pp. 258-259.
%F A255165 a(n) = Sum_{k=2..n} floor(log(n)/log(k)), n >= 1.
%F A255165 It appears that a(n) = A089361(n) + n - 1. - _Michel Marcus_, Feb 17 2015
%F A255165 From _Ridouane Oudra_, Nov 13 2019: (Start)
%F A255165 a(n) = Sum_{i=2..n} floor(n^(1/i)).
%F A255165 a(n) = Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1).
%F A255165 a(n) = A043000(n) - n + 1. (End)
%F A255165 a(n) ~ n. - _Vaclav Kotesovec_, Apr 06 2021
%e A255165 The first jump is at n = 4 where, in the summation, log(4)/log(2), as it reaches a new floor.
%e A255165 Note: Possible complications exist calculating the floor function on ratios of logs that produce exact integers (e.g., in Mathematica). Adding an infinitesimal amount to n solves it.
%t A255165 Table[Sum[Floor[Log[n]/Log[k]], {k, 2, n}], {n, 1, 100}]
%o A255165 (PARI) a(n)=sum(k=2,n,log(n)\log(k)) \\ _Anders Hellström_, Dec 06 2015
%o A255165 (Magma) [0] cat [&+[Floor(Log(n)/Log(k)):k in [2..n]]:n in [2..70]]; // _Marius A. Burtea_, Nov 13 2019
%Y A255165 Cf. A006218, A089361, A089723, A043000.
%K A255165 nonn,easy
%O A255165 1,3
%A A255165 _Richard R. Forberg_, Feb 15 2015