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Decimal expansion of Sum(1/p + 1/q) as (p, q) runs through the twin m^2 + 1 primes.
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%I #29 Oct 31 2022 02:06:13

%S 3,5,7,7,4,5,1,4,7

%N Decimal expansion of Sum(1/p + 1/q) as (p, q) runs through the twin m^2 + 1 primes.

%C Or decimal expansion of (1/5 + 1/17) + Sum_{i>=0} (1/p(i) + 1/q(i)) where p(i) and q(i) are primes of the form p(i) = m^2 + 1 = (10*i+4)^2 + 1 and q(i) = (m + 2)^2 + 1 = (10*i + 6)^2 + 1 (for m > 1, m == 4 (mod 10)). See A096012.

%C The sum is convergent; it must be less than 0.81459657... (see A172168).

%C Conjecture: the series of all twin m^2 + 1 prime reciprocals converges to 0.357745147...

%C It is probable that a(9) = 1.

%C A good approximation to the constant is (2*log(7/3)/log(17))^2 = 0.35774506... which agrees with the constant through the first 6 significant digits.

%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.

%D J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

%F Equals (1/5 + 1/17) + Sum_{n>=1} (1/(A096012(n)^2 + 1) + 1/(A096012(n) + 2)^2 + 1).

%e 0.3577451... = (1/5 + 1/17) + (1/17 + 1/37) + (1/197 + 1/257) + ...

%t s=N[1/5+1/17,20];Do[p=(10*k+4)^2+1;q=(10*k+6)^2+1;If[PrimeQ[p]&&PrimeQ[q],s=s+1/p+1/q],{k,0,10^7}];Print[N[s,20]]

%Y Cf. A002496, A005574, A065421, A085548, A096012, A172168, A206328.

%K nonn,cons,more

%O 0,1

%A _Michel Lagneau_, Jun 26 2018