A065493 -id:A065493 - cp's OEIS frontend

A078080 Continued fraction for constant defined in A065493.

0, 1, 1, 1, 20, 9, 1, 2, 5, 1, 2, 3, 2, 3, 38, 8, 1, 16, 2, 2, 21, 1, 12, 1, 2, 1, 1, 2, 43, 2, 1, 32, 10, 3, 221, 1, 2, 9, 1, 1, 3, 3, 52, 3, 4, 19, 1, 1, 1, 1, 5, 3, 3, 2, 1, 1, 1, 1, 3, 2, 2, 1, 8, 1, 1, 10, 5, 2, 4, 1, 1, 2, 2, 1, 8, 5, 3, 1, 2, 3, 1, 1, 4, 1, 4, 1, 2, 2, 3, 3, 5, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065493 (decimal expansion).

contfrac((1 + prodeulerrat(1 - 2/p^2))/2) \\ Amiram Eldar, Jul 08 2024

Offset changed and a(86) onwards corrected by Andrew Howroyd, Jul 05 2024

A065474 Decimal expansion of Product_{p prime} (1 - 2/p^2).

Original entry on oeis.org

3, 2, 2, 6, 3, 4, 0, 9, 8, 9, 3, 9, 2, 4, 4, 6, 7, 0, 5, 7, 9, 5, 3, 1, 6, 9, 2, 5, 4, 8, 2, 3, 7, 0, 6, 6, 5, 7, 0, 9, 5, 0, 5, 7, 9, 6, 6, 5, 8, 3, 2, 7, 0, 9, 9, 6, 1, 8, 1, 1, 2, 5, 2, 4, 5, 3, 2, 5, 0, 0, 6, 3, 4, 8, 6, 2, 4, 4, 6, 0, 9, 8, 8, 4, 5, 2, 3, 4, 8, 1, 5, 6, 8, 5, 6, 3, 7, 5, 5, 2, 1, 7, 7, 2, 7, 3

Offset: 0

N. J. A. Sloane, Nov 19 2001

Density of A007674, squarefree n such that n + 1 is squarefree. - Charles R Greathouse IV, Aug 10 2011
Product_{k>=1} (1 - 2/k^2) = sin(sqrt(2)*Pi) / (sqrt(2)*Pi). - Vaclav Kotesovec, May 23 2020
The asymptotic probability that, for two integers k and m, 0 < k <= m, we have gcd(k*(k+1), m) = 1 (when k and m are chosen at random in the range 1..n and n->oo) (Tóth and Sándor, 1989). - Amiram Eldar, Apr 29 2023

			0.322634098939244670579531692548...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Henri Cohen, High precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 163 and 184.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant F_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
L. Tóth and J. Sándor, An asymptotic formula concerning a generalized Euler function, Fibonacci Quarterly, Vol. 27, No. 2 (1989), pp. 176-180.
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Prime Products.
Zack Wolske, Number Fields with Large Minimal Index, Ph.D. Thesis, University of Toronto, 2018.

Cf. A007674, A058026, A065493, A074893.

$MaxExtraPrecision = 800; digits = 98; terms = 800; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{0, 2}, {-4, 0}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 2/p^2) \\ Amiram Eldar, Mar 16 2021

Edited by Dean Hickerson, Sep 10 2002

More digits from Vaclav Kotesovec, Dec 18 2019

A333634 Numbers with an even number of non-unitary prime divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102

Offset: 1

Amiram Eldar, May 23 2020

nonn

Numbers that have an even number of distinct prime factors raised to a power larger than 1.

The asymptotic density of this sequence is 0.661317... (A065493, Feller and Tornier, 1933).

			1 is a term since it has 0 prime divisors, and 0 is even.
180 is a term since 180 = 2^2 * 3^2 * 5 has 2 prime divisors, 2 and 3, with exponents larger than 1 in its prime factorization, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107 (1933), pp. 188-232.
I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Transactions of the American Mathematical Society, Vol. 39, No. 2 (1936), pp. 315-330. See p. 326.
Eric Weisstein's World of Mathematics, Feller-Tornier Constant.
Wikipedia, Feller-Tornier constant.

Cf. A056170, A065493, A190641, A327877 (complement).

Select[Range[100], EvenQ @ Count[FactorInteger[#][[;;,2]], u_ /; u > 1]  &]

Numbers k with A056170(k) == 0 (mod 2).

A327877 Numbers having an odd number of non-unitary prime factors.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168

Offset: 1

Amiram Eldar, Sep 28 2019

nonn

Differs from A190641(n) from n = 310 (900, the least number with 3 non-unitary prime factors, is in this sequence but not in A190641).

The asymptotic density of the numbers in this sequence is 0.338682... = 1 - A065493.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Transactions of the American Mathematical Society, Vol. 112, No. 2, (1964), pp. 214-227.
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107, No. 1 (1933), pp. 188-232.
I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Transactions of the American Mathematical Society, Vol. 39, No. 2 (1936), pp. 315-330.

Cf. A056170, A065493, A190641, A333634 (complement).

A056170[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Select[Range[200], OddQ[A056170[#]] &] (* after Jean-François Alcover at A056170 *)

\\ here b(n) is A056170(n)
b(n)={my(f=factor(n)[, 2]); sum(i=1, #f, f[i]>1)}
{ select(k->b(k)%2, [1..200]) } \\ Andrew Howroyd, Sep 28 2019

A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

Original entry on oeis.org

1, 0, 5, 6, 9, 3, 2, 2, 9, 1, 4

Offset: 1

Artur Jasinski, Jan 11 2021

Lesser twin primes A001359 (with the exception of the first prime, 3) are congruent to 5 mod 6: this constant is smaller than A340576.
By extrapolating method most probably the next two decimal digits are 1.056932291(46).
The known high-precision algorithms for Euler products are based on the Dirichlet L function and the Moebius inversion formula (see Mathematica procedure of Jean-François Alcover in A175646).
The constant is between 1.056932291453... and 1.056932291494. - R. J. Mathar, Feb 14 2025

			1.0569322914...

Cf. A005596, A065463, A065465, A065483, A065485, A065489, A065493, A072691, A082695, A111003, A175646.

One more digit confirmed by a bracketing of partial products - R. J. Mathar, Feb 14 2025

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A078080 Continued fraction for constant defined in A065493.

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A065474 Decimal expansion of Product_{p prime} (1 - 2/p^2).

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A333634 Numbers with an even number of non-unitary prime divisors.

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A327877 Numbers having an odd number of non-unitary prime factors.

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A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

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