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A306982
Decimal expansion of the upper asymptotic density of certain sets of residues.
1
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
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
KEYWORD
cons,nonn
AUTHOR
STATUS
approved