A342450 -id:A342450 - cp's OEIS frontend

A342451 a(n) is the denominator of the Schnirelmann density of the n-free numbers.

88, 189, 157, 3168, 6272, 236288, 587264, 2604717, 48833536, 293001216, 854822912, 12207734784, 8392914944, 183106404352, 1296997711872, 15258814251008, 2997264777216, 79472920887296, 556310058762240, 5960466160418816, 25033957873549312, 46938659244998656, 3099441608404238336

Offset: 2

Amiram Eldar, Mar 12 2021

See A342450 for details.

Amiram Eldar, Table of n, a(n) for n = 2..75 (from Hardy, 1979)
George Eugene Hardy, On the Schnirelmann density of the k-free and (k,r)-free integers, Ph.D. thesis, University of Alberta, 1979.

Cf. A342450 (numerators), A342452.

A342452 a(n) is the least number at which the Schnirelmann density of the n-free numbers is attained.

Original entry on oeis.org

176, 378, 2512, 3168, 31360, 236288, 1174528, 7814151, 48833536, 293001216, 1709645824, 12207734784, 67143319552

Offset: 2

Amiram Eldar, Mar 12 2021

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.
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.
Orr (1969) proved that 5^n <= a(n) < 6^n, for n >= 5.
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).
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.
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.

			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.

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.
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.

Cf. A005117, A342450, A342451.

A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 8, 3, 1, 2, 1, 6, 4, 1, 1, 2, 1, 2, 1, 1, 12, 1, 1, 4, 16, 10, 1, 1, 18, 8, 3, 1, 4, 1, 2, 5, 2, 27, 1, 2, 1, 6, 1, 32, 14, 3, 1, 1, 1, 2, 4, 1, 8, 25, 128, 1, 2, 9, 2, 4, 1, 2, 3, 1, 4, 2, 1, 8, 1, 2, 16, 1, 1, 2, 9, 1, 2, 6, 40, 4, 1, 2, 1

Offset: 1

Amiram Eldar, Jul 26 2022

f(n) = a(n)/A356094(n) is the asymptotic density of numbers k such that prime(n) = A053669(k) is the smallest prime not dividing k.

The asymptotic mean of A053669 is 2.9200509773... (A249270) which is the weighted mean of the primes with f(n) as weights. The corresponding asymptotic standard deviation, which can be evaluated from the second raw moment Sum_{n>=1} f(n) * prime(n)^2, is 2.8013781465... .

			Fractions begin with 1/2, 1/3, 2/15, 1/35, 1/231, 2/5005, 8/255255, 3/1616615, 1/10140585, 2/462120945, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002110, A006093, A039787, A053669, A249270, A356094 (denominators).

Similar sequences: A038110, A338559, A340818, A341431, A342450, A342479.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Numerator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 100}]]

a(n) = numerator((prime(n)-1)/factorback(primes(n))); \\ Michel Marcus, Jul 26 2022

from math import gcd
from sympy import prime, primorial
def A356093(n): return (p:=prime(n)-1)//gcd(p,primorial(n)) # Chai Wah Wu, Jul 26 2022

a(n) = 1 iff prime(n) is in A039787.
Let f(n) = a(n)/A356094(n):
f(n) = A006093(n)/A002110(n).
Sum_{n>=1} f(n) = 1.
Sum_{n>=1} f(n) * prime(n) = A249270.

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A342451 a(n) is the denominator of the Schnirelmann density of the n-free numbers.

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A342452 a(n) is the least number at which the Schnirelmann density of the n-free numbers is attained.

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A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

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