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Decimal expansion of Sum_{n>=1} 1/(prime(n)*prime(n+1)*prime(n+2)): Sum of reciprocals of products of three successive primes.
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%I #33 Apr 10 2024 09:13:48

%S 4,7,4,9,4,4,3,6,0,1,9,6,3,3,5,1,8,3,7

%N Decimal expansion of Sum_{n>=1} 1/(prime(n)*prime(n+1)*prime(n+2)): Sum of reciprocals of products of three successive primes.

%C Since p(n+1) > p(n) and p(n+2) > p(n), p(n)*p(n+1)*p(n+2) > p(n)^3 and 1/[p(n)*p(n+1)*p(n+2)] < 1/p(n)^3. Because Sum_{n>=1} 1/p(n)^3 = A085541 converges, Sum_{n>=1} 1/(p(n)*p(n+1)*p(n+2)) converges, too.

%C Greater than 0.04749443601963321719578... - _R. J. Mathar_, May 11 2014

%e 0.04749443601... = Sum_{n>=1} 1/A046301(n) = 1/(2*3*5) + 1/(3*5*7) + 1/(5*7*11) + 0.020286072... (primes 10 < p(n+1) < 100) + ...

%p proc(q) local n;

%p print(evalf(add(1/(ithprime(n)*ithprime(n+1)*ithprime(n+2)), n=1..q), 200));

%p end:

%Y Cf. A046301, A085541, A210473.

%K nonn,cons,more

%O -1,1

%A _Timothy Varghese_, May 06 2014

%E Offset corrected by _Jon E. Schoenfield_, Mar 21 2021

%E 9 more terms from _Jon E. Schoenfield_, Apr 10 2024