A336025 -id:A336025 - cp's OEIS frontend

A342450 a(n) is the numerator of the Schnirelmann density of the n-free numbers.

53, 157, 145, 3055, 6165, 234331, 584879, 2599496, 48785015, 292856489, 854612603, 12206236915, 8392400925, 183100803621, 1296977891119, 15258697717317, 2997253335821, 79472769236347, 556309528064071, 5960463317677243, 25033951904190895, 46938653648975843, 3099441423652148001

Offset: 2

Amiram Eldar, Mar 12 2021

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).

			The fractions begin with 53/88, 157/189, 145/157, 3055/3168, 6165/6272, 234331/236288, 584879/587264, 2599496/2604717, 48785015/48833536, 292856489/293001216, ...

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VI, p. 217.

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.
Wikipedia, Schnirelmann density.

Cf. A013928, A336025, A342451 (denominators), A342452.

Cf. A005117, A004709, A046100.

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).

A336026 Numbers m such that the proportion of nonsquarefree numbers in the interval [1, m] is greater than the corresponding proportion for all k > m.

Original entry on oeis.org

176, 380, 388, 389, 392, 393, 1089, 1864, 1928, 1936, 1937, 1940, 2080, 2892, 2900, 2908, 2909, 2912, 3776, 5589, 5788, 5832, 5932, 5933, 7156, 7157, 11881, 11889, 12656, 12776, 13880, 13888, 14085, 14088, 14096, 14104, 14456, 14464, 14465, 39740

Offset: 1

Javier Múgica, Jul 05 2020

nonn

Also, numbers m such that the proportion of squarefree numbers in the interval [1, m] is less than the corresponding proportion for all k > m.
If the condition "greater than" were changed to "greater than or equal to" the sequence would also include the number 396, where the proportion is the same as at 1089, namely, 156/396 = 429/1089 = 39/99. There seems to be no other such coincidence. There is none up to 2*10^11.
All the terms are congruent to 0 or 1 modulo 4. If the modulus 36 is considered, the only possible residue classes are 0, 1, 9, 12, 20, 28, 29, 32 and 33. Similar restrictions hold for larger moduli. Thus, mod 900 there are only 132 possible residues, the least one being 28. Of these, more than half appear in pairs of two consecutive values.

			There are 151 nonsquarefree numbers up to m = 380, for a proportion of 151/380 ~= 0.39737. This proportion is never again reached for larger values of m, so the number 380 belongs to this list.

Javier Múgica, Table of n, a(n) for n = 1..211
Javier Múgica, C program for finding (not proving) the terms of this sequence. The computations for the proofs can be found in the following spreadsheets: The first four terms. The first 211 terms. The determination of the possible residue classes mod 36 and 900.

Cf. A013929, A057627, A336025.

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A342450 a(n) is the numerator of the Schnirelmann density of the n-free numbers.

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