|
| |
|
|
A085967
|
|
Decimal expansion of the prime zeta function at 7.
|
|
9
|
|
|
|
0, 0, 8, 2, 8, 3, 8, 3, 2, 8, 5, 6, 1, 3, 3, 5, 9, 2, 5, 3, 5, 1, 2, 4, 1, 3, 8, 7, 2, 9, 4, 4, 8, 7, 2, 3, 0, 8, 9, 1, 8, 3, 3, 2, 8, 8, 8, 5, 3, 0, 7, 8, 0, 6, 2, 4, 4, 6, 4, 1, 9, 2, 1, 6, 3, 8, 6, 5, 5, 4, 8, 9, 4, 1, 1, 0, 0, 7, 8, 5, 8, 1, 8, 4, 3, 1, 6, 6, 1, 3, 4, 1, 8, 1, 9, 1, 8, 2, 0, 0, 4, 3, 2, 8, 1
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 07 2017
|
|
|
REFERENCES
|
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
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..1902
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
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(7) = Sum_{p prime} 1/p^7 = Sum_{n>=1} mobius(n)*log(zeta(7*n))/n.
Equals Sum_{k>=1} 1/A092759(k). - Amiram Eldar, Jul 27 2020
|
|
|
EXAMPLE
|
0.0082838328561335925351...
|
|
|
MAPLE
|
A085967:= proc(i) print(evalf(add(1/ithprime(k)^7, k=1..i), 100)); end:
A085967(100000); # Paolo P. Lava, May 29 2012
|
|
|
MATHEMATICA
|
s[n_] := s[n] = Join[{0, 0}, Sum[ MoebiusMu[k]*Log[Zeta[7*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104] & // First]; s[100]; s[n = 200]; While[ s[n] != s[n - 100], n = n + 100]; s[n] (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[ PrimeZetaP[ 7], 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(7, 47)*10^105)));
// Jason Kimberley, Dec 30 2016
(PARI) sumeulerrat(1/p, 7) \\ Hugo Pfoertner, Feb 03 2020
|
|
|
CROSSREFS
|
Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085966 (at 6), this sequence (at 7), A085968 (at 8), A085969 (at 9).
Cf. A013665, A092759.
Sequence in context: A021551 A143025 A303326 * A163960 A246849 A143531
Adjacent sequences: A085964 A085965 A085966 * A085968 A085969 A085970
|
|
|
KEYWORD
|
cons,easy,nonn
|
|
|
AUTHOR
|
Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
|
|
|
STATUS
|
approved
|
| |
|
|
