A306982 Decimal expansion of the upper asymptotic density of certain sets of residues.

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

Offset

-8,2

Links

Robert G. Wilson v, Table of n, a(n) for n = -8..10000

Carl Pomerance, Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.

Formula

1 - 6/Pi^2 Sum_{j=0..5} delta_j, where delta_0 = 1 and delta_j = (1/j)* Sum_{i=1..j} (-1)^(i-1) eta_i delta_{j-i}.

Example

0.0000000155377352117679639022335262675814992572444423241569849361213...
Sum_{j=0..5} delta_j = 1.64493404128967646496804948687389729523...

Mathematica

digits = 105;
delta[1] = eta[1] = N[Sum[PrimeZetaP[2n], {n, 1, 4 digits}], digits];
eta[2] = N[Sum[n PrimeZetaP[2n+2], {n, 1, 4 digits}] , digits];
eta[3] = N[Sum[n(n+1)/2 PrimeZetaP[2n+4], {n, 1, 4 digits}], digits];
eta[4] = N[Sum[n(n+1)(n+2)/6 PrimeZetaP[2n+6], {n, 1, 4 digits}], digits];
eta[5] = N[Sum[n(n-1)(n-2)(n-3)/24 PrimeZetaP[2n+2], {n, 1, 4 digits}], digits];
delta[0]=1; delta[j_] := 1/j Sum[(-1)^(i-1) eta[i] delta[j-i], {i, 1, j}];
d = 1 - 6/Pi^2 Sum[delta[j], {j, 0, 5}];
RealDigits[d][[1]]

Crossrefs

Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

Sequence in context: A195693 A232813 A267033 * A278928 A273826 A213054

Adjacent sequences: A306979 A306980 A306981 * A306983 A306984 A306985

Keyword

cons,nonn

Author

Jean-François Alcover, Mar 18 2019

Status

approved