
COMMENTS

This sequence has the maximal length of a powerful arithmetic progression for which the kth term is a kth power.
The originating sequence is 1, 9, 17, 25, 33 with difference 8. This sequence is multiplied by 3^24*5^30*11^24*17^20 to generate a(n) with common difference 84237040758000422943661278378274517566101748109131201632359035313129425048828125000.
Note that this sequence is just an example of a maximal progression. Similar progressions with smaller terms are provided by 2^15*3^24*5^40*13^24 * {11, 18, 25, 32, 39}, 37^24 * {213, 169, 125, 81, 37}, or, if negative terms are allowed, by 2^15*5^20 * {11, 8, 5, 2, 1}.  Giovanni Resta, Aug 29 2017


EXAMPLE

a(1) is obviously a first power.
a(2) = 307841957589849138828884412917083740234375^2 is a square.
a(3) = 5635779747116948576103515625^3 is a third power.
a(4) = 716288998461106640625^4 is a fourth power.
a(5) = 51072299355515625^5 is a fifth power.
