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A210473 Decimal expansion of Sum_{n>=1} 1/(prime(n)*prime(n+1)). 4
3, 0, 1, 0, 9, 3, 1, 7, 6, 3, 5, 8, 3, 9, 9, 8, 9, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Sum of reciprocals of products of successive primes. Differs from A209329 only by the initial term 1/(2*3) = 1/6 = 0.16666...

Being p(n+1)>p(n) then p(n)*p(n+1)>p(n)^2 and 1/[p(n)*p(n+1)]<1/p(n)^2. Because of the convergence of Sum_{n>=1}1/p(n)^2 (A085548) even Sum_{n>=1}1/[p(n)*p(n+1)] converges. - Paolo P. Lava, May 21 2013

LINKS

Table of n, a(n) for n=0..17.

FORMULA

Equals 1/6 + A209329.

EXAMPLE

0.3010931763... = sum( 1/(prime(n)*prime(n+1)), n=1,2,3...)

= 1/(2*3) + 1/(3*5) + 1/(5*7)

+ 0.03731790933454338 (primes 10 < p(n+1) < 100)

+ 0.0017430141479028 (primes 100 < p(n+1) < 10^3)

+ 0.00011767024549033 (primes 10^3 < p(n+1) < 10^4)

+ 9.018426684045269 e-6 (primes 10^4 < p(n+1) < 10^5)

+ 7.3452282601302 e-7 (primes 10^5 < p(n+1) < 10^6)

+ 6.19161299373 e-8 (primes 10^6 < p(n+1) < 10^7)

+ 5.3439026467 e-9 (primes 10^7 < p(n+1) < 10^8)

+ 4.70035656 e-10 (primes 10^8 < p(n+1) < 10^9) + ...

MAPLE

A210473:=proc(q) local n;

print(evalf(add(1/(ithprime(n)*ithprime(n+1)), n=1..q), 200));

end: A210473(10^6); # Paolo P. Lava, May 21 2013

MATHEMATICA

digits = 10;

f[n_Integer] := 1/(Prime[n]*Prime[n+1]);

s = NSum[f[n], {n, 1, Infinity}, Method -> "WynnEpsilon", NSumTerms -> 2*10^6, WorkingPrecision -> MachinePrecision];

RealDigits[s, 10, digits][[1]] (* Jean-Fran├žois Alcover, Sep 05 2017 *)

PROG

(PARI) S(L=10^9, start=3)={my(s=0, q=1/precprime(start)); forprime(p=1/q+1, L, s+=q*q=1./p); s} \\ Using 1./p is maybe a little less precise, but using s=0. and 1/p takes about 50% more time.

(PARI) {my( tee(x)=printf("%g, ", x); x ); t=vector(8, n, tee(S(10^(n+1), 10^n))); s=1/2/3+1/3/5+1/5/7; vector(#t, n, s+=t[n])} \\ Shows contribution of sums over (n+1)-digit primes (vector t) and the vector of partial sums; the final value is in s.

CROSSREFS

Cf. A085548, A209329, A185380.

Sequence in context: A128311 A132884 A319234 * A185951 A188832 A279514

Adjacent sequences:  A210470 A210471 A210472 * A210474 A210475 A210476

KEYWORD

nonn,cons

AUTHOR

M. F. Hasler, Jan 23 2013

EXTENSIONS

Corrected and extended by Hans Havermann, Mar 17 2013 using the additional terms of A209329 from R. J. Mathar, Feb 08 2013

STATUS

approved

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Last modified November 18 21:04 EST 2018. Contains 317331 sequences. (Running on oeis4.)