Postage Function, Using [ ]

If the postal rate for first class letters is 42 cents for the first ounce or any portion thereof, and 17 cents for every ounce or portion thereof after the first ounce, then find the cost of a letter weighing w ounces, where w is some positive real number. Use [x] to stand for the greatest integer in w, as usual.

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Progression to Quadratic

Consider the following interesting list of integers:

P0 = 41,
P1 = 43 = 41 + 2,
P2 = 47 = 43 + 4,
P3 = 53 = 47 + 6, … ,

in which Pn is obtained by adding 2n to Pn-1.

It so happens that there is a quadratic function F(x) with the property that Pn = F(n) for all nonnegative n. Find a formula for F(x).

Once you have the formula, consider the question of whether or not F(n) is a prime number for every nonnegative value of n.

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