[seqfan] Re: A164771 Average digit of n^2 is s=1.

Meeussen Wouter (bkarnd) wouter.meeussen at vandemoortele.com
Wed Aug 26 13:55:09 CEST 2009

in the same spirit, but with the integer digits in base b,
where the average random digit would be (b-1)/2:

4095 is the only integer n below 10^5 for which the average of the
integer digits
of n^2 in base b equals (b-1)/2 for all even bases [2,4,6,8,10]
(but fails at b=12, 14, 16...)

below 10^6, there is just also 955290 with that property
(also fails at b=12,14, but ok at b=16).

On the odd-base-side,
n<10^5, these 33 make it for bases [3,5,7,9] but not 11
{1554, 1590, 12354, 12486, 13422, 14334, 14538, 14586, 14754, 15054,
45744,
46020, 46344, 47340, 50172, 51132, 52008, 52176, 52896, 53772, 53952,
54528,
55344, 55512, 56964, 57048, 57336, 57360, 57636, 58068, 58152, 58176,
58560}

below n=10^6, no takers for bases [3,5,7,9,11].
So 11 is not a nice base.

Is that because all n,
such that average-of-base-11-digits of n^2 equal 5,
are multiples of 5,
and the increasing list of the n/5
consists of runs with equal parity like
1 even, 3 odd, 6 even, 15 odd, 45 even, 110 odd, 413,1399,4244,13495,...
Now thats stuff for inclusion in a new BookOfNumbers-version!
;-))

Wouter.

--- toxic code below, beware ----
it=Select[Range[1000000],Apply[Plus,#]/Length[#]&
[IntegerDigits[#^2,11]]===5 &];
Length/@ Split[it/5,Mod[#1,2]===Mod[#2,2]&]
gives {1, 3, 6, 15, 45, 110, 413, 1399, 4244, 13495, 15350} (*last
doesn't count*)
---------------------------------

-----Original Message-----
From: seqfan-bounces at list.seqfan.eu
[mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of zak seidov
Sent: woensdag 26 augustus 2009 11:16
To: seqfan at seqfan.eu
Subject: [seqfan] A164771 Average digit of n^2 is s=1.

Just submitted
A164771
1,1049,1490,10002,10005,10011,10020,10050,10101,10110,10149,10200,10500,
11001,11010,11100,11490,12000,14499,15000,17610,18000,20001,20010,20100,
21000,24900,30000,33200,35000,36100,44900,44990,45100
There are 117 such n's < 10^7.

Is it worth submitting also cases s=3..9?
Thx,
Zak

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