A316189 - OEIS

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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, 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 = (10i+4)^2 + 1 and q(i) = (m + 2)^2 + 1 = (10i + 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=(10k+4)^2+1; q=(10k+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.)