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 A342450 #17 Jan 05 2025 19:51:41
%S A342450 53,157,145,3055,6165,234331,584879,2599496,48785015,292856489,
%T A342450 854612603,12206236915,8392400925,183100803621,1296977891119,
%U A342450 15258697717317,2997253335821,79472769236347,556309528064071,5960463317677243,25033951904190895,46938653648975843,3099441423652148001
%N A342450 a(n) is the numerator of the Schnirelmann density of the n-free numbers.
%C A342450 k-free numbers are numbers whose exponents in their prime factorization are all less than k. E.g., the squarefree numbers (k=2, A005117), the cubefree numbers (k=3, A004709) and the biquadratefree numbers (k=4, A046100).
%C A342450 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.
%C A342450 a(2) was found by Rogers (1964).
%C A342450 a(3)-a(6) were found by Orr (1969).
%C A342450 a(7)-a(75) were found by Hardy (1979).
%D A342450 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VI, p. 217.
%H A342450 Amiram Eldar, <a href="/A342450/b342450.txt">Table of n, a(n) for n = 2..75</a> (from Hardy, 1979)
%H A342450 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 A342450 R. L. Duncan, <a href="https://doi.org/10.1090/S0002-9939-1965-0186652-1">The Schnirelmann density of the k-free integers</a>, Proceedings of the American Mathematical Society, Vol. 16, No. 5 (1965), pp. 1090-1091.
%H A342450 R. L. Duncan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-2/duncan.pdf">On the density of the k-free integers</a>, Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 140-142.
%H A342450 Paul Erdős, G. E. Hardy and M. V. Subbarao, <a href="https://users.renyi.hu/~p_erdos/1978-32.pdf">On the Schnirelmann density of k-free integers</a>, Indian J. Math., Vol. 20 (1978), pp. 45-56.
%H A342450 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 A342450 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.
%H A342450 Kenneth Rogers, <a href="https://doi.org/10.1090/S0002-9939-1964-0163893-X">The Schnirelmann density of the squarefree integers</a>, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.
%H A342450 Harold M. Stark, <a href="https://doi.org/10.1090/S0002-9939-1966-0199161-1">On the asymptotic density of the k-free integers</a>, Proceedings of the American Mathematical Society, Vol. 17, No. 5 (1966), pp. 1211-1214.
%H A342450 M. V. Subbarao, <a href="https://core.ac.uk/download/pdf/39229699.pdf">On the Schnirelman density of the K-free integers</a>, Distribution of values of arithmetic functions, Vol. 517 (1984), pp. 47-61; <a href="https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/98401">alternative link</a>.
%H A342450 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SchnirelmannDensity.html">Schnirelmann Density</a>.
%H A342450 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schnirelmann_density">Schnirelmann density</a>.
%F A342450 Let d(n) = a(n)/A342451(n), and let D(n) = 1/zeta(n), the asymptotic density of the n-free numbers. Then:
%F A342450 Lim_{n->oo} d(n) = 1.
%F A342450 d(n) < D(n) (Stark, 1966).
%F A342450 d(n) < D(n) < d(n+1) < D(n+1) (Duncan, 1965; Erdős et al., 1978).
%F A342450 d(n) > 1 - Sum_{p prime} 1/p^n (Duncan, 1969).
%F A342450 (D(n+1)-d(n+1))/(D(n)-d(n)) < 1/2^n (Duncan, 1969).
%F A342450 d(n) > 1 - 1/2^n - 1/3^n - 1/5^n (Diananda and Subbarao, 1977).
%e A342450 The fractions begin with 53/88, 157/189, 145/157, 3055/3168, 6165/6272, 234331/236288, 584879/587264, 2599496/2604717, 48785015/48833536, 292856489/293001216, ...
%Y A342450 Cf. A013928, A336025, A342451 (denominators), A342452.
%Y A342450 Cf. A005117, A004709, A046100.
%K A342450 nonn,frac
%O A342450 2,1
%A A342450 _Amiram Eldar_, Mar 12 2021