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A319459
Decimal expansion of the minimum of the error term E(x) in exp(x) * x^(-x) * Pi^(-1/2) * Gamma(1 + x) = (8*x^3 + 4*x^2 + x + E(x))^(1/6) for x > 0. The location where the minimum is achieved is provided in A319458.
1
1, 0, 0, 4, 5, 0, 7, 1, 8, 7, 7, 4, 8, 4, 5, 1, 9, 6, 8, 8, 4, 8, 3, 6, 7, 5, 2, 1, 6, 8, 9, 9, 1, 0, 6, 5, 1, 3, 6, 0, 5, 1, 3, 4, 1, 3, 2, 4, 6, 1, 9, 8, 3, 7, 9, 8, 2, 4, 7, 7, 5, 1, 1, 0, 0, 3, 5, 3, 2, 4, 0, 3, 5, 1, 7, 4, 1, 5, 7, 7, 9, 0, 6, 9, 9, 8, 4, 9, 2
OFFSET
-1,4
COMMENTS
Ramanujan's question 754 in the Journal of the Indian Mathematical Society (VIII, 80) asked "Show that exp(x) * x^(-x) * Pi^(-1/2) * Gamma(1 + x) = (8*x^3 + 4*x^2 + x + E)^(1/6), where E lies between 1/100 and 1/30 for all positive values of x".
A numerical search provides an approximate minimum of E = 0.010045071877... at x = 0.6715... (A319458), confirming Ramanujan's lower bound.
REFERENCES
See A319458.
EXAMPLE
0.0100450718774845196884836752168991065136051341...
CROSSREFS
Cf. A319458.
Sequence in context: A322505 A192041 A132022 * A318740 A240160 A249860
KEYWORD
nonn,cons
AUTHOR
Hugo Pfoertner, Sep 19 2018
STATUS
approved