A316189 Decimal expansion of Sum(1/p + 1/q) as (p, q) runs through the twin m^2 + 1 primes.

3, 5, 7, 7, 4, 5, 1, 4, 7

Offset: 0

Author

Michel Lagneau, Jun 26 2018

Keywords

• nonn
• cons
• more

Comments

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.

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

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

It is probable that a(9) = 1.

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.

Examples

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

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.
• J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Crossrefs

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

Programs

Mathematica

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]]

Formula

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