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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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equals the Dirichlet zeta-function 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. 329-345.

Robert E. Dressler, A density which counts multiplicity, Pacific J. Math. 34 (1970), pp. 371-378.

P. Erdős, Some remarks on Euler's ϕ function and some related problems, Bull. Amer. Math. Soc. 51 (1945), pp. 540-544.

J. von Zur Gathen et al., Average order in cyclic groups, J. Theor. Nombres Bordeaux, 16 (2004), 107-123. Lists several other papers where this constant arises.

S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.

D. Handelman, Invariants for critical dimension groups and permutation-Hermite 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/(p-1), 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,changed

AUTHOR

Benoit Cloitre, Apr 12 2003

EXTENSIONS

New definition from Eric W. Weisstein, May 04 2006

STATUS

approved

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Last modified November 23 09:20 EST 2014. Contains 249840 sequences.