[seqfan] Re: A055198 comment seems wrong

[M. F. Hasler]

I agree,
I also confirm "The maximum for 111 is 99885" rather than 99889.
(PARI)
is_A055198(n,U=[n],i=n)=until(setsearch(U,n=A004086(n+4)),U=setunion(U,[n]));n==i
(with A004086(n)=fromdigits(Vecrev(digits(n)))  = "reverse(n)")
shows that the next terms of oeis.org/A055198 should be:
select( is_A055198, [1000..1999])
[1011, 1013, 1015, 1017, 1019, 1021, 1023, 1025, 1027, 1029, 1031, 1033,
1035, 1037, 1039, 1041, 1043, 1045, 1047, 1049, 1051, 1053, 1055, 1057,
1059, 1061, 1063, 1065, 1067, 1069, 1071, 1073, 1075, 1077, 1079, 1081,
1083, 1085, 1087, 1089, 1091, 1093, 1095, 1097, 1121, 1123, 1125, 1127,
1129, 1131, 1133, 1135, 1137, 1139, 1141, 1143, 1145, 1147, 1149, 1151,
1153, 1155, 1157, 1159, 1161, 1163, 1165, 1167, ...]
The respective minimum elements of the cycles (which is a natural
representative of that cycle) are
 [1011, 1013, 1015, 1017, 1019, 1021, 1023, 1025, 1027, 1029, 1031, 1033,
1035, 1037, 1039, 1041, 1043, 1045, 1047, 1049, 1051, 1053, 1053, 1051,
1049, 1047, 1045, 1043, 1041, 1039, 1037, 1035, 1033, 1031, 1029, 1027,
1025, 1023, 1021, 1019, 1017, 1015, 1013, 1011, 1011, 1013, 1015, 1017,
1019, 1021....]
This shows that there are 22 new cycles here, respectively represented by
all odd numbers from 1011 to 1053.
The next cycle contains only numbers larger than 10000.
The number 10111 appears to have a huge trajectory... It quickly reaches
7-digit numbers :
a(2)= 51101, a(4)= 91105, a(44)= 99185, a(495)= 99999, a(497)= 500003,
a(499)= 900007,
a(544)= 990097, a(944)= 998897, a(994)= 999997, a(995) = 1000001
and then oscillates there, e.g. the terms from a(1000) on are :
3100013, 7100013, 7100017, 1200017, 1200021, 5200021, 5200025, 9200025,
9200029, 3300029, 3300033, 7300033, 7300037, 1400037, 1400041, 5400041,
5400045, 9400045, 9400049, 3500049, 3500053, 7500053, 7500057, 1600057,
1600061, 5600061, 5600065, 9600065, 9600069, 3700069, 3700073, 7700073,
7700077, 1800077, 1800081, 5800081, 5800085, 9800085, 9800089, 3900089,
3900093, 7900093, 7900097, 1010097, 1010101, ...
5000 terms later it is again at 3100213 ...
The orbit has a size of (much?) more than 100 000 terms, but up to there
the size of terms doesn't increase beyond 7 digit numbers.
Some more record values are:
a(1039) = 98 000 89,  a(1489) = 98 909 89, a(3489) = 98 989 89,
a(5989) = 98 990 89, a(8489) = 98 991 89, a(10989) = 98 992 89, a(13489) =
98 993 89,
a(15989) = 98 994 89,  a(18489) = 98 995 89,  a(20989) = 98 996 89,
a(26 038) = 99 000 85, a(26388) = 997 0 785, a(26438)= 998 0 885,
a(26488) = 999 0 985, a(26988)= 999 2 985, a(27488)= 999 4 985,
a(27988) = 999 6 985, a(28488) = 999 8 985,
then no more records for a while...

- Maximilian

On Tue, Mar 15, 2022 at 7:00 PM Hans Havermann <gladhobo at bell.net> wrote:

> "all integers <= 10000 eventually iterate to a cycle based on these 54,
> although for example 111 takes 2635 iterations to do so via a maximum of
> 99889"
>
> The maximum for 111 is 99885. I'm inclined to agree with Sean that the 54
> are just the tip of the iceberg. Here is 1011:
>
> 1011, 5101, 5015, 9105, 9019, 3209, 3123, 7213, 7127, 1317, 1231, 5321,
> 5235, 9325, 9239, 3429, 3343, 7433, 7347, 1537, 1451, 5541, 5455, 9545,
> 9459, 3649, 3563, 7653, 7567, 1757, 1671, 5761, 5675, 9765, 9679, 3869,
> 3783, 7873, 7787, 1977, 1891, 5981, 5895, 9985, 9899, 3099, 3013, 7103,
> 7017, 1207, 1121, 5211, 5125, 9215, 9129, 3319, 3233, 7323, 7237, 1427,
> 1341, 5431, 5345, 9435, 9349, 3539, 3453, 7543, 7457, 1647, 1561, 5651,
> 5565, 9655, 9569, 3759, 3673, 7763, 7677, 1867, 1781, 5871, 5785, 9875,
> 9789, 3979, 3893, 7983, 7897, 1097, 1011, ...
>
>
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