A365901 - OEIS

%I #9 Oct 12 2023 17:50:17

%S 1,2,1,4,2,1,6,1,12,2,1,24,4,2,1,36,6,1,48,8,4,2,1,60,2,1,120,4,2,1,

%T 180,6,1,240,8,4,2,1,360,12,2,1,720,24,4,2,1,840,4,2,1,1260,6,1,1680,

%U 8,4,2,1,2520,12,2,1,5040,24,4,2,1,7560,36,6,1

%N Irregular triangle read by rows giving trajectory beginning with A002182(n) under recursion of x -> f(x) until reaching 1, where f(x) = x/rad(x), rad(x) = A007947(x).

%C Let h(n) = A002182(n).

%C Since highly composite numbers h(n) are products of primorials (i.e., in A025487), the squarefree kernel is always a primorial (i.e., in A002110), and the trajectory always reaches 1.

%H Michael De Vlieger, <a href="/A365901/b365901.txt">Table of n, a(n) for n = 1..11397</a> (rows n = 1..1200, flattened)

%H Michael De Vlieger, <a href="/A365901/a365901.png">Log log scatterplot of log_10(a(n))</a>, n = 1..1053678, showing 2^16 rows.

%F Length of row n is A112779(n)+1.

%F T(n,2) = A301413(n), n > 1.

%e Row 1 = {1} since h(1) = 1, already 1.

%e Row 2 = {2, 1} since h(2) = 2, 2/rad(2) = 2/2 = 1, reaching 1.

%e Row 3 = {4, 2, 1} since h(3) = 4, 4/rad(4) = 4/2 = 2, and we follow the trajectory of 2 thereafter.

%e Row 6 = {24, 4, 2, 1} since h(6) = 24, 24/rad(24) = 24/6 = 4, and we follow the trajectory of 4 thereafter.

%e Row 14 = {720, 24, 4, 2, 1} since h(14) = 720, 720/rad(720) = 720/30 = 24, which appends row 6 thereafter.

%e .

%e First rows of this sequence:

%e row 1:   1

%e     2:   2, 1

%e     3:   4, 2, 1

%e     4:   6, 1

%e     5:  12, 2, 1

%e     6:  24, 4, 2, 1

%e     7:  36, 6, 1

%e     8:  48, 8, 4, 2, 1

%e     9:  60, 2, 1

%e    10: 120, 4, 2, 1

%e    11: 180, 6, 1

%e    12: 240, 8, 4, 2, 1

%e    ...

%t a2182 = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]]; Array[NestWhileList[#/(Times @@ FactorInteger[#][[All, 1]]) &, a2182[[#]], # > 1 &] &, 20] // TableForm

%Y Cf. A002110, A002182, A007947, A112779, A301413, A365900.

%K nonn,tabf

%O 1,2

%A _Michael De Vlieger_, Oct 11 2023