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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.
e = 2/prime(0)# + 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 number.
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allocated for Albert LauPrimorial 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
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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.
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.
e = 2/prime(0)# + 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 number.
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]
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Albert Lau, Apr 06 2014
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allocated for Albert Lau
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