[seqfan] Re: n and n^2 use digits 0, 1, 4, 6 only: A136859 - a question

[David Wilson]

I doubt it's a theorem.
Let 0, 1, 4, and 6 be the good digits, and let number n be good if n includes only those digits.
If n starts with 0, then n = 0, and trivially n and n^2 are both good.
I believe I can prove that if n and n^2 are both good, and n starts with 1, then n = 10^k.
I believe I can also prove that if n and n^2 are both good, and n starts with 4, then n = 4*10^k.

But when I tried the same proof for n starting with 6, I ran into a snag.
I now believe that there are good numbers n starting with 6 such that n^2 starts with an arbitrarily large number of good digits.
If true, this would imply that that there is real n on [0.6, 0.7) such that n and n^2 are both good.
I'm pretty convinced that infinitely many such n exist.
The smallest such n, to 1000 decimals, is almost certainly

n =
0.640401440041010640140411100106000466101016140014400001040140400060110610160061
16041464616014146460106000100660606644104141100004060000160411141100146600140016
01644001414411000144646400040660006000060061611600414061141160006060406000014010
60000104060600104006600664116644106016010410646601401440000101614640016041100604
00141400410041406046060600600614100014060141060640001000600144406100460016101440
66040046061406606004600146666000011060040646000616601061000000644111001110110411
14611661040164011000400110106664401114001000101616144041141440641661001406614011
66401000014000661114141140001011041140111400640116046014016164401001660066606116
60401166010040000064041410041411040104464600044061000101411610010646061606141010
00011600664161040044001000640060014046640004040400001060461600666006000611016006
44411001660066466601660140116011004010041406406066006146014101116006110101401004
66041411040110116616416466000061400110446110060610600064116100114661060000040061
161464060614464466464000014010464601400041

With

n^2 = 
0.410114004406600146004164014004460000440141414000106000011604000006640001416600
01406061001040006000641101606004141601400014441406444064444644404406016100406640
64601404101161006041606041616404661141611614464104601610010000441004100161401004
11611041606161400161441600041606440100600104144040060011161104100004100014101441
04660166060614044414600106114000006646444004014010604100101604164640144401100004
00604404010106040044041604600401144000401010116140110416000466000400404441440104
00110004400100404164040040416014100400616166614001016404010004004004040010161664
00006404401614614040410011106164404414040116011640604601640400064104001060104000
44010460100014441060604110610064401640101140440610016046600164141601614441416164
04440460001044414600466006001044440600010141404014640046000066101661610140111410
04104044406116060044014416644464000044060040000440600001600440400041441046040100
60006600001641000141106001441404146640006006446000016401641614400600400044616644
100161014040060016060640601044100144004010

I doubt it is rational.

Press any key to continue . . .
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> Sloane
> Sent: Friday, January 29, 2016 6:36 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] n and n^2 use digits 0,1,4,6 only: A136859 - a question
> 
> A136859 contains the numbers n such that n and n^2 use only the digits
> 0,1,4,6.
> 
> It appears that n must be 0, 10^k, or 4*10^k. Does anyone know if this is a
> theorem?
> 
> I don't know how far it has been checked.
> 
> The entry had a formula which was based on there being no other type of
> term, but this was only a conjecture.  I deleted the program and the b-file
> that were based on the conjecture.
> 
> Thanks to Maximilian for raising suspicions about the formula.
> 
> It would be nice if someone could check that the terms shown are complete,
> that there are no missing numbers. The terms shown are the original ones
> found by Jonathan Wellons, so they are probably correct, but it would be
> good to have a check.
> 
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