A240455 Primorial expansion of Pi.

3, 0, 0, 4, 1, 8, 1, 0, 8, 19, 13, 10, 28, 29, 23, 30, 9, 32, 4, 26, 12, 27, 75, 28, 45, 30, 47, 65, 91, 83, 9, 92, 123, 44, 73, 32, 140, 102, 28, 75, 108, 30, 139, 4, 127, 88, 57, 182, 207, 172, 80, 126, 150, 232, 227, 19, 256, 238, 195, 44, 56, 58, 131, 160, 243, 222, 22, 47, 30, 226, 312, 130, 161, 68, 358, 52, 250, 152, 15, 38, 120, 195, 120, 263, 412, 115, 412, 427, 284, 361, 121, 413, 355, 75, 473, 355, 10, 177, 101, 71

Offset

0,1

Comments

The primorial expansion a(n) of a real number x is defined as x = 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.

Index entries for sequences related to primorial base

Index entries for sequences related to the number Pi

Formula

x(0) = Pi; a(n) = floor(x(n)) where x(n + 1) = prime(n + 1) * (x(n) - a(n)) and prime(n) = A000040(n) is the n-th prime number. [corrected by Rémy Sigrist, Jan 06 2019]

Example

Pi = 3/prime(0)# + 0/prime(1)# + 0/prime(2)# + 4/prime(3)# + 1/prime(4)# + 8/prime(5)# + ... where prime(n)# = A002110(n) is the n-th primorial number. [

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[\[Pi], 100]

Crossrefs

Cf. A000796 (decimal expansion), A075874 (factorial number system expansion).

Sequence in context: A364084 A288654 A259191 * A304263 A305223 A316801

Adjacent sequences: A240452 A240453 A240454 * A240456 A240457 A240458

Keyword

nonn,base

Author

Albert Lau, Apr 05 2014

Status

approved