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 A342450 a(n) is the numerator of the Schnirelmann density of the n-free numbers. 4
 53, 157, 145, 3055, 6165, 234331, 584879, 2599496, 48785015, 292856489, 854612603, 12206236915, 8392400925, 183100803621, 1296977891119, 15258697717317, 2997253335821, 79472769236347, 556309528064071, 5960463317677243, 25033951904190895, 46938653648975843, 3099441423652148001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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). 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. a(2) was found by Rogers (1964). a(3)-a(6) were found by Orr (1969). a(7)-a(75) were found by Hardy (1979). REFERENCES József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VI, p. 217. LINKS Amiram Eldar, Table of n, a(n) for n = 2..75 (from Hardy, 1979) P. H. Diananda and M. V. Subbarao, On the Schnirelmann density of the k-free integers, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (1977), pp. 7-10. R. L. Duncan, The Schnirelmann density of the k-free integers, Proceedings of the American Mathematical Society, Vol. 16, No. 5 (1965), pp. 1090-1091. R. L. Duncan, On the density of the k-free integers, Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 140-142. Paul Erdős, G. E. Hardy and M. V. Subbarao, On the Schnirelmann density of k-free integers, Indian J. Math., Vol. 20 (1978), pp. 45-56. George Eugene Hardy, On the Schnirelmann density of the k-free and (k,r)-free integers, Ph.D. thesis, University of Alberta, 1979. Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319. Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516. Harold M. Stark, On the asymptotic density of the k-free integers, Proceedings of the American Mathematical Society, Vol. 17, No. 5 (1966), pp. 1211-1214. M. V. Subbarao, On the Schnirelman density of the K-free integers, Distribution of values of arithmetic functions, Vol. 517 (1984), pp. 47-61; alternative link. Eric Weisstein's World of Mathematics, Schnirelmann Density. Wikiepdia, Schnirelmann density. FORMULA Let d(n) = a(n)/A342451(n), and let D(n) = 1/zeta(n), the asymptotic density of the n-free numbers. Then: Lim_{n->oo} d(n) = 1. d(n) < D(n) (Stark, 1966). d(n) < D(n) < d(n+1) < D(n+1) (Duncan, 1965; Erdős et al., 1978). d(n) > 1 - Sum_{p prime} 1/p^n (Duncan, 1969). (D(n+1)-d(n+1))/(D(n)-d(n)) < 1/2^n (Duncan, 1969). d(n) > 1 - 1/2^n - 1/3^n - 1/5^n (Diananda and Subbarao, 1977). EXAMPLE The fractions begin with 53/88, 157/189, 145/157, 3055/3168, 6165/6272, 234331/236288, 584879/587264, 2599496/2604717, 48785015/48833536, 292856489/293001216, ... CROSSREFS Cf. A013928, A336025, A342451 (denominators), A342452. Cf. A005117, A004709, A046100. Sequence in context: A217718 A044385 A044766 * A160058 A353136 A053070 Adjacent sequences: A342447 A342448 A342449 * A342451 A342452 A342453 KEYWORD nonn,frac AUTHOR Amiram Eldar, Mar 12 2021 STATUS approved

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