

A082695


Decimal expansion of zeta(2)*zeta(3)/zeta(6).


9



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

1,2


COMMENTS

Equals the Dirichlet zetafunction sum_{n>=1} A001615(n)/n^s at s=3.  R. J. Mathar, Apr 02 2011
Dressler shows that this is the average value of A014197, that is, the number of values m such that phi(m) <= n is asymptotically n times this constant. Erdős had shown earlier that this limit exists.  Charles R Greathouse IV, Nov 26 2013
From Stanislav Sykora, Nov 14 2014: (Start)
Equals lim(n>infinity)(sum(k=1..n,k/phi(k))/n), i.e., the limit mean value of k/phi(k), where phi(k) is the Euler's totient function.
Equals also lim(n>infinity)(sum(k=1..n,1/phi(k))/log(n)).
Proofs are trivial using the formulas for sum(k=1..n,k/phi(k)) and sum(k=1..n,1/phi(k)) listed in the Wikipedia link.
For the limit mean value of phi(k)/k, see A059956. (End)


REFERENCES

Joe Roberts, Lure of the Integers, Mathematical Association of America, 1992. See p. 74.


LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Paul T. Bateman, The distribution of values of the Euler function, Acta Arithmetica 21:1 (1972), pp. 329345.
Robert E. Dressler, A density which counts multiplicity, Pacific J. Math. 34 (1970), pp. 371378.
P. Erdős, Some remarks on Euler's ϕ function and some related problems, Bull. Amer. Math. Soc. 51 (1945), pp. 540544.
J. von Zur Gathen et al., Average order in cyclic groups, J. Theor. Nombres Bordeaux, 16 (2004), 107123. Lists several other papers where this constant arises.
S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848852.
D. Handelman, Invariants for critical dimension groups and permutationHermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013.
Eric Weisstein's World of Mathematics, Totient Summatory Function
Eric Weisstein's World of Mathematics, Powerful Number
Wikipedia, Euler's totient function


FORMULA

Decimal expansion of product( 1+1/p/(p1), for all prime p) = zeta(2)*zeta(3)/zeta(6)=1.94359643682075920505707...
The sum of the reciprocals of the powerful numbers, A001694.  T. D. Noe, May 03 2006
Equals A013661 * A002117 / A013664 = 1 / A068468 = zeta(3) * 315/(2*Pi^4) = zeta(3) * A157292.


EXAMPLE

1.94359643682075920505707036257476343718785850176780571602663568890 ...


MATHEMATICA

First@RealDigits[ Zeta[2]*Zeta[3]/Zeta[6], 10, 100]
RealDigits[ 315 Zeta[3]/(2 Pi^4), 10, 111][[1]] (* Robert G. Wilson v, Aug 11 2014 *)


PROG

(PARI) zeta(3)*315/2/Pi^4


CROSSREFS

Cf. A014197, A059956, A070243, A082696 (continued fraction).
Sequence in context: A021110 A010540 A187466 * A236257 A019909 A227324
Adjacent sequences: A082692 A082693 A082694 * A082696 A082697 A082698


KEYWORD

cons,nonn


AUTHOR

Benoit Cloitre, Apr 12 2003


EXTENSIONS

New definition from Eric W. Weisstein, May 04 2006


STATUS

approved



