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Numbers k such that the numerator of Sum_{i=1..k} 1/prime(i), in reduced form, is prime.
7

%I #23 Sep 16 2024 12:01:00

%S 2,3,5,6,18,19,22,47,57,58,63,70,73,112,632,1382,4621

%N Numbers k such that the numerator of Sum_{i=1..k} 1/prime(i), in reduced form, is prime.

%C Terms <= 112 correspond to certified primes.

%C Numbers k such that the arithmetic derivative of the k-th primorial, A024451(k) [= A003415(A002110(k))] is prime. - _Antti Karttunen_, Jan 09 2024

%e Sum_{i=1..6} 1/prime(i) = 40361/30030 and 40361 is prime, hence 6 is a term.

%t s = 0; Do[s += 1/Prime[n]; k = Numerator[s]; If[PrimeQ[k], Print[n]], {n, 1, 1500}]

%t Position[Accumulate[1/Prime[Range[120]]],_?(PrimeQ[ Numerator[ #]]&)] //Flatten (* To generate terms greater than 120, increase the Range constant, but the program may take much longer to run. *) (* _Harvey P. Dale_, Jan 01 2019 *)

%Y Cf. A002110, A003415.

%Y Positions of primes in A024451.

%K hard,more,nonn

%O 1,1

%A _Ryan Propper_, Aug 02 2005

%E a(17) from _Michael S. Branicky_, Sep 16 2024