A335131 -id:A335131 - cp's OEIS frontend

A206256 Decimal expansion of Product_{p prime} (1 - 3/p^2).

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

Offset: 0

N. J. A. Sloane, Feb 05 2012, based on a posting by Warren Smith to the Math Fun Mailing List, Feb 04 2012

For a randomly selected number k, this is the probability that k, k+1, k+2 all are squarefree.

			0.1254869809058...

Leon Mirsky, Note on an asymptotic formula connected with r-free integers, The Quarterly Journal of Mathematics, Vol. os-18, No. 1 (1947), pp. 178-182.
Leon Mirsky, Arithmetical pattern problems relating to divisibility by rth powers, Proceedings of the London Mathematical Society, Vol. s2-50, No. 1 (1949), pp. 497-508.

Cf. A059956, A065474, A007675, A335131, A370600.

# See A175640 using efact := 1-3/p^2. - R. J. Mathar, Mar 22 2012

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 3}, {0, -6}, m]; RealDigits[(1/4) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Oct 01 2019 *)

prodeulerrat(1 - 3/p^2) \\ Amiram Eldar, Mar 16 2021

More terms from Amiram Eldar, Oct 01 2019

More terms from Vaclav Kotesovec, Dec 17 2019

A330319 a(n) = Sum_{i=1..n} phi(i)*phi(i+1), where phi(n) = A000010(n) is Euler's totient function.

Original entry on oeis.org

1, 3, 7, 15, 23, 35, 59, 83, 107, 147, 187, 235, 307, 355, 419, 547, 643, 751, 895, 991, 1111, 1331, 1507, 1667, 1907, 2123, 2339, 2675, 2899, 3139, 3619, 3939, 4259, 4643, 4931, 5363, 6011, 6443, 6827, 7467, 7947, 8451, 9291, 9771, 10299, 11311, 12047, 12719, 13559, 14199, 14967, 16215, 17151, 17871, 18831

Offset: 1

N. J. A. Sloane, Dec 11 2019

nonn

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 32.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.
L. Mirsky, Summation formula involving arithmetic functions, Duke Mathematical Journal, Vol. 16, No. 2 (1949), pp. 261-272.

Cf. A000010, A065474, A335131.

Partial sums of A083542.

phi = EulerPhi[Range[56]]; Accumulate[Most[phi] * Rest[phi]] (* Amiram Eldar, Mar 05 2020 *)

a(n) = sum(i=1, n, eulerphi(i)*eulerphi(i+1)); \\ Michel Marcus, Mar 05 2020

a(n) ~ (c/3) * n^3 + O(n^2*log(n)^2), where c = Product_{p prime}(1 - 2/p^2) (A065474). - Amiram Eldar, Mar 05 2020

cp's OEIS Frontend

A206256 Decimal expansion of Product_{p prime} (1 - 3/p^2).

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