|
| |
|
|
A085965
|
|
Decimal expansion of the prime zeta function at 5.
|
|
6
|
|
|
|
0, 3, 5, 7, 5, 5, 0, 1, 7, 4, 8, 3, 9, 2, 4, 2, 5, 7, 1, 3, 2, 8, 1, 8, 2, 4, 2, 5, 3, 8, 8, 5, 5, 7, 1, 1, 1, 3, 1, 6, 9, 7, 2, 7, 6, 7, 2, 6, 6, 5, 1, 3, 3, 1, 6, 9, 0, 0, 9, 2, 6, 7, 4, 8, 2, 3, 9, 7, 5, 8, 3, 4, 2, 7, 4, 7, 2, 7, 9, 3, 1, 3, 6, 6, 0, 7, 2, 8, 0, 6, 4, 7, 0, 3, 7, 6, 7, 9, 5, 0, 8, 9, 6, 3, 9
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017
|
|
|
REFERENCES
|
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
|
|
|
LINKS
|
Jason Kimberley, Table of n, a(n) for n = 0..1702
H. Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint.
X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Eric Weisstein's World of Mathematics, Prime Zeta Function
|
|
|
FORMULA
|
P(5) = Sum_{p prime>=2} 1/p^5 = Sum_{n>=1} mobius(n)*log(zeta(5*n))/n.
Equals 1/2^5 +A085994 +A086035. - R. J. Mathar, Jul 14 2012
|
|
|
EXAMPLE
|
0.0357550174839242571328...
|
|
|
MAPLE
|
A085965:= proc(i) print(evalf(add(1/ithprime(k)^5, k=1..i), 100)); end:
A085965(100000); # Paolo P. Lava, May 29 2012
|
|
|
MATHEMATICA
|
s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[5*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n=200]; While[s[n] != s[n-100], n = n+100]; s[n] (* Jean-François Alcover, Feb 14 2013, from 1st formula *)
RealDigits[ PrimeZetaP[ 5], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)
|
|
|
PROG
|
(MAGMA) R := RealField(106);
PrimeZeta := func<k, N | &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)): n in[1..N]]>;
[0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(5, 69)*10^105)));
// Jason Kimberley, Dec 30 2016
|
|
|
CROSSREFS
|
Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4), this sequence (at 5), A085966 (at 6) to A085969 (at 9).
Cf. A013663.
Sequence in context: A023899 A279321 A254863 * A238205 A186702 A141710
Adjacent sequences: A085962 A085963 A085964 * A085966 A085967 A085968
|
|
|
KEYWORD
|
cons,easy,nonn
|
|
|
AUTHOR
|
Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
|
|
|
STATUS
|
approved
|
| |
|
|
