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 A342452 #6 Mar 12 2021 23:51:43
%S A342452 176,378,2512,3168,31360,236288,1174528,7814151,48833536,293001216,
%T A342452 1709645824,12207734784,67143319552
%N A342452 a(n) is the least number at which the Schnirelmann density of the n-free numbers is attained.
%C A342452 k-free numbers are numbers whose exponents in their prime factorization are all less than k. Let Q_k(m) be the number of k-free numbers not exceeding m. The Schnirelmann density for k-free numbers is d(k) = inf_{m>=1} Q_k(m)/m. See A342450 for more information.
%C A342452 The value of m(k) in which Q_k(m)/m = d(k) is not necessarily unique: while for k = 2, 3 and 4 the density is attained at a single value, i.e., 176, 378 and 2512, respectively, for k = 5 the density is attained at both 3168 and 6336. Hardy (1979) found that also for k = 38, 55 and 56 the value of m(k) is not unique, and for k = 38 the density is attained in at least 3 values.
%C A342452 Orr (1969) proved that 5^n <= a(n) < 6^n, for n >= 5.
%C A342452 Diananda and Subbarao (1977) proved that the largest value of m at which the density is attained is in the interval [6^n/2, 6^n).
%C A342452 Hardy (1969) calculated the least value of m in this interval, for n = 2..75, but his values are not necessarily the least nor the largest.
%C A342452 The terms in the data section for n=2..14 were verified to be the least values. Except for n=5, they are also unique values.
%H A342452 P. H. Diananda and M. V. Subbarao, <a href="https://doi.org/10.1090/S0002-9939-1977-0435024-9">On the Schnirelmann density of the k-free integers</a>, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (1977), pp. 7-10.
%H A342452 George Eugene Hardy, <a href="https://archive.org/details/Hardy1979">On the Schnirelmann density of the k-free and (k,r)-free integers</a>, Ph.D. thesis, University of Alberta, 1979.
%H A342452 Richard C. Orr, <a href="https://doi.org/10.1112/jlms/s1-44.1.313">On the Schnirelmann density of the sequence of k-free integers</a>, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.
%e A342452 The number of squarefree numbers (A005117) up to 176 is Q_2(176) = 106. It is where the Schnirelmann density inf_{m>=1} Q_2(m)/m = 106/176 = 53/88 is attained. Therefore a(2) = 176.
%Y A342452 Cf. A005117, A342450, A342451.
%K A342452 nonn,more
%O A342452 2,1
%A A342452 _Amiram Eldar_, Mar 12 2021