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A316189 Decimal expansion of Sum(1/p + 1/q) as (p, q) runs through the twin m^2 + 1 primes. 0
3, 5, 7, 7, 4, 5, 1, 4, 7, 1, 4, 1, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

REFERENCES

S. 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.

LINKS

Table of n, a(n) for n=0..12.

FORMULA

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

EXAMPLE

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

From Jon E. Schoenfield, Sep 03 2018: (Start)

Let s(k) be the partial sum of the reciprocals of the twin m^2 + 1 primes less than (10*2^k)^2, and let t(k) = s(k) - s(k-1). Then using s'(k) = s(k) + t(k)^2/(t(k-1) - t(k)) to accelerate the convergence gives

.

   k          s(k)                 s'(k)

  ==  ====================  ====================

  ...

  20  0.357745146822891...  0.357745147131985...

  21  0.357745146993709...  0.357745147138589...

  22  0.357745147073285...  0.357745147142687...

  23  0.357745147109860...  0.357745147140970...

  24  0.357745147126658...  0.357745147140924...

  25  0.357745147134491...  0.357745147141338...

  26  0.357745147138151...  0.357745147141359...

  27  0.357745147139859...  0.357745147141353...

  28  0.357745147140658...  0.357745147141361...

  29  0.357745147141033...  0.357745147141364...

  30  0.357745147141209...  0.357745147141365... (End)

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]]

CROSSREFS

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

Sequence in context: A127314 A294499 A092257 * A342668 A297346 A342622

Adjacent sequences:  A316186 A316187 A316188 * A316190 A316191 A316192

KEYWORD

nonn,cons,more

AUTHOR

Michel Lagneau, Jun 26 2018

EXTENSIONS

a(9)-a(12) from Jon E. Schoenfield, Jul 14 2018

STATUS

approved

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Last modified May 25 16:32 EDT 2022. Contains 354071 sequences. (Running on oeis4.)