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Sequence of the Day for March 19

A119523: Decimal expansion of the van der Waerden–Ulam binary measure of the primes.

0.829365019702...
The formula is

k  = 1
χp(k)
2k  − 1
, where
χp(n)
is the characteristic function of the primes. The primes have a larger measure than the composites because they dominate the smaller integers.

The binary expansion of the van der Waerden–Ulam binary measure of the primes being obviously (see A010051)

0.1101010001010001010001000001010000010001010001...

The quaternary (base 4) expansion of the van der Waerden–Ulam binary measure of the primes being (see A??????)

0.31101101101001100101101...

In the quaternary expansion all the digits after the point, with the exception of the initial 3, are 0 or 1 since there is only one even prime.



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