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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... Since prime(n+1) > prime(n), we have prime(n)*prime(n+1) > prime(n)^2 and 1/(prime(n)*prime(n+1)) < 1/prime(n)^2. Because of the convergence of Sum_{n>=1} 1/prime(n)^2 (A085548), Sum_{n>=1} 1/(prime(n)*prime(n+1)) also converges. - Paolo P. Lava, May 21 2013 LINKS FORMULA Equals 1/6 + A209329. EXAMPLE 0.3010931763... = Sum_{n>=1} 1/(prime(n)*prime(n+1)). = 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-François 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: A334076 A132884 A319234 * A185951 A188832 A279514 Adjacent sequences:  A210470 A210471 A210472 * A210474 A210475 A210476 KEYWORD nonn,cons,more 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 May 28 17:37 EDT 2020. Contains 334684 sequences. (Running on oeis4.)