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Powerful numbers (A001694) with a record gap to the next powerful number.
6

%I #15 Mar 14 2024 11:13:08

%S 1,4,9,16,36,49,81,144,169,256,289,441,529,576,676,729,900,1024,1156,

%T 1225,1372,1444,1600,1849,2209,2401,2916,3600,3721,4096,4356,4624,

%U 4761,5041,5625,6400,6561,7225,7396,8281,9025,9409,9801,11025,11236,11664,12544,14400

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

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

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

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

%H Amiram Eldar, <a href="/A349062/b349062.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%e 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.

%t 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]

%Y Cf. A001694, A060355, A076445, A371191.

%K nonn

%O 1,2

%A _Amiram Eldar_, Nov 07 2021