A349062 Powerful numbers (A001694) with a record gap to the next powerful number.

1, 4, 9, 16, 36, 49, 81, 144, 169, 256, 289, 441, 529, 576, 676, 729, 900, 1024, 1156, 1225, 1372, 1444, 1600, 1849, 2209, 2401, 2916, 3600, 3721, 4096, 4356, 4624, 4761, 5041, 5625, 6400, 6561, 7225, 7396, 8281, 9025, 9409, 9801, 11025, 11236, 11664, 12544, 14400

Offset: 1

Author

Amiram Eldar, Nov 07 2021

Keywords

• nonn

Comments

This sequence is infinite since the asymptotic density of the powerful numbers is 0.

The corresponding record gaps are 3, 4, 7, 9, 13, 15, 19, 25, 27, 32, 35, 43, ...

Apparently, most of the terms are squares. The nonsquare terms are 1372, 465125, 4879688, ... (A371191).

Examples

			The sequence of powerful numbers begins with 1, 4, 8, 9, 16, 25, 27, 32, 36 and 49. The differences between these terms are 3, 4, 1, 7, 9, 2, 5, 4 and 13. The record values, 3, 4, 7, 9 and 13 occur after the powerful numbers 1, 4, 9, 16 and 36, the first 5 terms of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to powerful numbers.

Crossrefs

Cf. A001694, A060355, A076445, A371191.

Programs

Mathematica

powQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 1; seq[nmax_] := Module[{s = {}, n1 = 1, gapmax = 0, gap}, Do[If[powQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, nmax}]; s]; seq[10^5]