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

Author

Albert Lau, Apr 05 2014

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

• Index entries for sequences related to primorial base
• Index entries for sequences related to the number Pi

Crossrefs

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

Programs

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]

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]