A256391 -id:A256391 - cp's OEIS frontend

A256392 Decimal expansion of Product_{p prime} (1-4/p^2+4/p^3-1/p^4).

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

Offset: 0

Juan Arias-de-Reyna, Mar 28 2015

Also decimal expansion of the probability that an integer tuple (x,y,z,w) satisfies gcd(x,y) = gcd(y,z) = gcd(z,w) = gcd(w,x) = 1.

			0.2177787166195363783230075141...

Juan Arias-de-Reyna, Table of n, a(n) for n = 0..1000
Juan Arias de Reyna and Randell Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq., Vol. 18 (2015), Article 15.10.4; arXiv preprint, arXiv:1403.2769 [math.NT], 2014.

Cf. A065473, A126775, A196524, A256391, A330523, A336643.

Do[Print[N[Exp[-Sum[q = Expand[(4 p^2 - 4 p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 50]], {t, 10, 100, 10}] (* Vaclav Kotesovec, Dec 17 2019 *)

prodeulerrat(1-4/p^2+4/p^3-1/p^4) \\ Amiram Eldar, Mar 03 2021

A256390 a(n) = number of triples (a,b,c) of natural numbers a,b,c <= n with gcd(a,b)=gcd(b,c)=gcd(c,a)=1.

Original entry on oeis.org

1, 4, 13, 22, 55, 64, 133, 172, 247, 280, 469, 508, 781, 868, 997, 1144, 1621, 1714, 2323, 2488, 2785, 3010, 3907, 4078, 4837, 5176, 5833, 6178, 7627, 7798, 9463, 10102, 10927, 11530, 12631, 13006, 15379, 16150, 17311, 17926, 20863, 21256

Offset: 1

Juan Arias-de-Reyna, Mar 27 2015

nonn

The sequence has asymptotics rho*n^3+O(n^2 log^2n) with rho=prod_p(1-3/p^2+2/p^3)=0.2867474284344...(product on primes). See A065473.

			a(3)=13 because the 13 triples (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,1,3), (1,3,1), (3,1,1), (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..1399
J. Arias de Reyna and R. Heyman, Counting tuples restricted by pairwise primality, arXiv:1403.2769 [math.NT], 2014.
J. Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4

Cf. A256391.

A[M_] := A[M] = Module[{X, a1, a2, a3, K, count, k},
    X = Flatten[
      Table[{a1, a2, a3}, {a1, 1, M}, {a2, 1, M}, {a3, 1, M}], 2];
    K = Length[X];
    count = 0;
    For[k = 1, k <= K, k++,
     {a1, a2, a3} = X[[k]];
     If[(GCD[a1, a2] == 1) && (GCD[a2, a3] == 1) && (GCD[a3, a1] ==
         1), count = count + 1]];
    count];
Table[A[n], {n, 1, 100}]

a(n) = sum_a sum_b sum_c mu(a) mu(b) mu(c) [n/gcd(a,b)][n/gcd(b,c)][n/gcd(c,a)], where mu(.) is Moebius function [x] integer part of x, and a,b,c run through natural numbers.

cp's OEIS Frontend

A256392 Decimal expansion of Product_{p prime} (1-4/p^2+4/p^3-1/p^4).

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