Repeating Digits in the Squares of Integers

Today I turned 25197 days old and none of the entries in the Online Encyclopaedia of Integer Sequences (OEIS) made much sense to me. I searched about for something of interest about this number and, after some dead-ends, I thought I'd square the number. The result was 634888809 which immediately caught my attention because of the repetition of the digit 8. I thought this result would be relatively rare and something that distinguishes 25197 from most other numbers. My question was how rare? I set about investigating using a Google spreadsheet and had soon found all the numbers between 1 and 26000 whose squares contain at least four consecutive 8's. It turns out that there is a total of only 14 such numbers.

Here are the numbers with their associated squares

Runs of four or more 8's (there are 14 of these):

1609	2588881
6992	48888064
9428	88887184
10094	101888836
12202	148888804
16090	258888100
16667	277788889
16849	283888801
20221	408888841
20359	414488881
21187	448888969
22917	525188889
24267	588887289
25197	634888809

It can be seen that the squares of 12202 and 20221 contain a run of five 8's. Naturally, I was curious about runs of the other digits 1, 2, 3, 4, 5, 6, 7 and 9 and once again used the spreadsheet to investigate the matter. The results are detailed below:

Run of four or more 1's (there are 11 such numbers:

2848	8111104
3334	11115556
10541	111112681
10542	111133764
10543	111154849
10544	111175936
10545	111197025
17062	291111844
20276	411116176
20521	421111441
23877	570111129

What's interesting above is the cluster of numbers 10541, 10542, 10543, 10544 and 10545 that all produce at least four leading 1's (10541 produces five leading 1's).

Runs of four or more 2's (there are 18 of these):

3335	11122225
3496	12222016
4585	21022225
4714	22221796
5415	29322225
6101	37222201
8332	69422224
8557	73222249
13335	177822225
14585	212722225
15415	237622225
16665	277722225
16668	277822224
20548	422220304
20838	434222244
23335	544522225
24585	604422225
25415	645922225

Runs of four or more 3's (there are eight of these):

5774	33339076
7303	53333809
11547	133333209
18037	325333369
18257	333318049
18258	333354564
18259	333391081
23094	533332836

There is another cluster here consisting of 18257, 18258 and 18259, all producing a run of four leading 3's when squared. There is also a run of five 3's produced by 11547 when it is squared.

Runs of four or more 4's (there are 12 of these):

5696	32444416
6667	44448889
11595	134444025
15393	236944449
15857	251444449
21081	444408561
21082	444450724
21083	444492889
24229	587044441
24788	614444944
25386	644448996
25771	664144441

Here again there is a cluster of 21081, 21082 and 21083, all producing a run of four 4's when squared. One number 15857 produces a run of five 4's when squared. The square of 24788 contains six 4's though only four of them are consecutive.

Runs of four or more 5's (there are 14 of these):

2357	5555449
3944	15555136
4166	17355556
8334	69455556
10274	105555076
10419	108555561
12472	155550784
15986	255552196
16666	277755556
20629	425555641
20834	434055556
22925	525555625
23570	555544900
23571	555592041

Here we see that the square of 22925 contains six 5's, although only four of them are consecutive.

Runs of four or more 6's (there are ten of these):

1291	1666681
1633	2666689
2582	6666724
5164	26666896
8165	66667225
12910	166668100
16330	266668900
17224	296666176
25819	666620761
25820	666672400

Here there are a pair of numbers, 25819 and 25820, that both produce a run of four leading 6's when squared.

Runs of four or more 7's (there are seven of these):

8819	77774761
11076	122677776
13924	193877776
16964	287777296
18915	357777225
21858	477772164
24037	577777369

Runs of four or more 9's (there are only two of these):

7071	49999041
14142	199996164

It's surprising that there are only two numbers between 1 and 26000 that produce four sequential 9's when squared and the one is double the other. I'll need to investigate the reason for this paucity.

I've proposed the sequence of numbers that produce four or more 8's when squared as a candidate for the OEIS. If it's approved, I'll propose the other runs of four or more digits are candidate sequences as well.