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

1,2

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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Example

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.

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]

Crossrefs

Cf. A001694, A060355, A076445, A371191.

Sequence in context: A068952 A000548 A256944 * A106575 A025620 A117218

Adjacent sequences: A349059 A349060 A349061 * A349063 A349064 A349065

Keyword

nonn

Author

Amiram Eldar, Nov 07 2021

Status

approved