A240472 Primorial expansion of e.

2, 1, 1, 1, 3, 9, 3, 0, 1, 1, 16, 25, 8, 3, 32, 32, 37, 24, 53, 17, 28, 67, 52, 2, 21, 81, 56, 88, 9, 3, 80, 42, 15, 37, 107, 52, 32, 120, 49, 46, 84, 3, 129, 29, 159, 103, 90, 172, 128, 98, 202, 138, 207, 150, 249, 131, 132, 66, 9, 86, 137, 191, 236, 141, 222, 285, 8, 205, 310, 250, 63, 173, 288, 93, 294, 84, 66, 104, 28, 154, 93, 229, 96, 254, 333, 89, 126, 393, 388, 396, 418, 424, 356, 299, 482, 64, 114, 60, 513, 471

Offset

0,1

Comments

The primorial expansion a(n) of a real number x is defined as x = a(0) + sum(i>0, a(i) / prime(i)# ) where a(0) = floor(x) and 0 <= a(i) < prime(i) for all i > 0.

Links

Table of n, a(n) for n=0..99.

Formula

x(0) = e;

a(n) = floor(x(n));

x(n + 1) = prime(n) * (x(n) - a(n));

where prime(n) = A000040(n) is the n-th prime number.

a(n) gives the primorial expansion of x(0) = e.

Example

e = 2 + 1/prime(1)# + 1/prime(2)# + 1/prime(3)# + 3/prime(4)# + 9/prime(5)# + ...
where prime(n)# = A002110(n) is the n-th primorial.

Mathematica

pe = Block[{x = #, $MaxExtraPrecision = \[Infinity]},
       Do[x = Prime[i] (x - Sow[x // Floor]) // Expand, {i, #2 - 1}];
       x // Floor // Sow] // Reap // Last // Last // Function;
pe[E, 100]

Crossrefs

Sequence in context: A251660 A279453 A054252 * A366836 A007442 A362483

Adjacent sequences: A240469 A240470 A240471 * A240473 A240474 A240475

Keyword

nonn

Author

Albert Lau, Apr 06 2014

Status

approved