Home > Numbers
96 Tees (Posted on 2023-11-01) |
|
A "Tee" is a decimal digit that includes the letter "t", or simply " two, three or eigh t".
What is the smallest positive integer N such that CONCAT(N^1,N^2, ... ,N^7,N^8), i.e. the concatenation of the first 8 powers of N includes 96 "Tees".
No Solution Yet
|
Submitted by Larry
|
No Rating
|
|
computer solution
|
| Comment 1 of 2
|
clearvars for n= sym(50): sym(3000000) c=[char(string(n)) ... char(string(n^2)) ... char(string(n^3)) ... char(string(n^4)) ... char(string(n^5)) ... char(string(n^6)) ... char(string(n^7)) ... char(string(n^8)) ... ]; ct=length(find(c=='2'|c=='3'|c=='8')); if ct==96 disp([n ct]) end end
finds first (and once found, I stopped the program):
[1186022, 96]
so the answer is 1,186,022.
The powers are
1186022, 1406648184484, 1668315693058082648, 1978659114912133298346256, 2346733240786318158771223233632, 2783277251703870635302163721998691904, 3301028052620328058622342821892332569365888, 3915091893024866724743388278306388058584469217536
The concatenation is 224 digits long.
In the vicinity of 1186022, the 96 occurrences is a high outlier. It's the only one that's 96 between 1186000 and 1187000. The first 50 counts in this area are
[1186000, 28] [1186001, 69] [1186002, 63] [1186003, 77] [1186004, 70] [1186005, 71] [1186006, 68] [1186007, 66] [1186008, 71] [1186009, 67] [1186010, 52] [1186011, 61] [1186012, 80] [1186013, 59] [1186014, 63] [1186015, 75] [1186016, 69] [1186017, 62] [1186018, 77] [1186019, 51] [1186020, 67] [1186021, 67] [1186022, 96] <-- the only 96 [1186023, 77] [1186024, 64] [1186025, 73] [1186026, 69] [1186027, 71] [1186028, 62] [1186029, 67] [1186030, 73] [1186031, 72] [1186032, 78] [1186033, 80] [1186034, 66] [1186035, 67] [1186036, 66] [1186037, 80] [1186038, 73] [1186039, 74] [1186040, 55] [1186041, 74] [1186042, 67] [1186043, 59] [1186044, 62] [1186045, 67] [1186046, 68] [1186047, 65] [1186048, 66] [1186049, 58] [1186050, 62]
in keeping with what we might expect as 224 * .3 = 67.2
In contrast a different range of one thousand, higer up by an order of magnitude:
[21002023, 96] [21002152, 96] [21002221, 96] [21002467, 96] [21002823, 96] [21002935, 96]
where the lengths are of the order of 267, and 267 * .3 = 80.1.
or
[31002131, 96] [31002206, 96] [31002207, 96] [31002233, 96] [31002265, 96] [31002428, 96] [31002532, 96] [31002536, 96] [31002631, 96] [31002688, 96] [31002787, 96] [31002808, 96] [31002853, 96] [31002911, 96] [31002984, 96]
where the concatenations are closer to 272 long and 272 * .3 = 81.6.
or to this span of 1000:
[331002031, 96] [331002033, 96] [331002047, 96] [331002049, 96] [331002056, 96] [331002061, 96] [331002064, 96] [331002081, 96] [331002120, 96] [331002121, 96] [331002225, 96] [331002262, 96] [331002338, 96] [331002363, 96] [331002424, 96] [331002468, 96] [331002536, 96] [331002537, 96] [331002552, 96] [331002625, 96] [331002628, 96] [331002669, 96] [331002724, 96] [331002756, 96] [331002763, 96] [331002782, 96] [331002802, 96] [331002807, 96] [331002814, 96] [331002839, 96] [331002852, 96] [331002867, 96] [331002896, 96]
where the concatenated lengths are over 310.
I'm sure that as the numbers get higher, eventually 96 will become an outlier as lower than the average altogether.
|
Posted by Charlie
on 2023-11-01 15:52:56 |
|
|
Please log in:
Forums (0)
Newest Problems
Random Problem
FAQ |
About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On
Chatterbox:
|