A291042 One powerful arithmetic progression with nontrivial difference and maximal length.

10529630094750052867957659797284314695762718513641400204044879414141178131103515625, 94766670852750475811618938175558832261864466622772601836403914727270603179931640625, 179003711610750898755280216553833349827966214731903803468762950040400028228759765625, 263240752368751321698941494932107867394067962841035005101121985353529453277587890625, 347477793126751744642602773310382384960169710950166206733481020666658878326416015625

Offset: 1

Author

Martin Renner, Aug 16 2017

Keywords

• nonn
• fini
• full

Comments

This sequence has the maximal length of a powerful arithmetic progression for which the k-th term is a k-th 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

Examples

			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.
		

Links

• John P. Robertson, The maximal length of a powerful arithmetic progression, American Mathematical Monthly 107 (2000), 951.

Crossrefs

Cf. A050923.