A277587 Decimal expansion of the sum of the reciprocals of the Wieferich primes.

0, 0, 1, 1, 9, 9, 7, 3, 2, 2, 2, 3, 1, 0, 3, 2, 8, 8

Offset

0,5

Comments

A heuristic argument predicts about log(log(x)) Wieferich primes below some number x (see for example Dorais, Klyve, 2011, result 14). PrimeGrid has searched to about 2.7 * 10^18 as of Apr 03 2021 without finding a third Wieferich prime, so the constant is known to 18 places after the decimal point.

It seems that if there are only finitely many Wieferich primes, then the constant is rational and if there are infinitely many Wieferich primes, then the constant is irrational.

The period lengths of 1/1093 and 1/3511 are 1092 and 3510, respectively (cf. Garza, Young, 2004).

Links

Table of n, a(n) for n=0..17.

F. G. Dorais and D. Klyve, A Wieferich Prime Search up to 6.7 × 10^15, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.2.

G. Garza and J. Young, Wieferich Primes and Period Lengths for the Expansions of Fractions, Mathematics Magazine, Vol. 77, No. 4 (2004), 314-319.

PrimeGrid, Subproject status (See "Wieferich and Wall-Sun-Sun Prime Search" at the bottom).

Example

1/1093 + 1/3511 = 4604/3837523 ~ 0.001199732223103288...

Crossrefs

Cf. A001220.

Sequence in context: A197390 A298520 A270712 * A372828 A201570 A019895

Adjacent sequences: A277584 A277585 A277586 * A277588 A277589 A277590

Keyword

nonn,cons,hard,more

Author

Felix Fröhlich, Nov 22 2016

Extensions

a(12)-a(17) from Felix Fröhlich, Apr 03 2021

Status

approved