# [seqfan] Re: Sum of a(n) first digits of S is a(n+1)

Jason Kimberley Jason.Kimberley at newcastle.edu.au
Fri Jan 8 01:02:25 CET 2010

Hello seqfans,

In the interests of my own sleep I have also been restricting my attention to the constant solutions of this problem.

By brute force, for about 10 minutes, I get these constant solutions:
0,1,11,20,102,111,120,201,210,300,1012,1021,1102,1111,1120,1201,1300,2002,2011,2020,2200,3100,4000,10031,10040,10121,10130,10211,10220,10301,10310,10400,11012,11021,11030,11102,11111,11120,11201,11210,11300,12011,12020,12101,12110,12200,13001,13010,13100,14000,20012,20030,20102,20120,20210,20300,21020,21110,21200,22010,22100,23000,30020,30110,30200,31010,31100,32000,40010,40100,41000,

We can weaken the condition  "the terms of A061384 which have the additional property of being a multiple of their sum-of-digits" to be "the terms k of A061384 with d digits such that the first (k mod d) digits of k have digit sum equal to (k mod d)". Perhaps someone can easily prove that this is a sufficient condition for the constant solution [k,k,...] to be a solution to the original problem "Sum of a(n) first digits of A is a(n+1)". (We could consider 0 to be the empty string "" with no digits.)

The solutions to my "k mod d" condition up to 7 digits are below.

Thanks,
Jason.

[1, 11, 20, 102, 111, 120, 201, 210, 300, 1012, 1021, 1102, 1111, 1120, 1201,
1300, 2002, 2011, 2020, 2200, 3100, 4000, 10031, 10040, 10121, 10130, 10211,
10220, 10301, 10310, 10400, 11012, 11021, 11030, 11102, 11111, 11120, 11201,
11210, 11300, 12011, 12020, 12101, 12110, 12200, 13001, 13010, 13100, 14000,
20012, 20030, 20102, 20120, 20210, 20300, 21020, 21110, 21200, 22010, 22100,
23000, 30020, 30110, 30200, 31010, 31100, 32000, 40010, 40100, 41000, 50000,
100014, 100032, 100050, 100104, 100122, 100140, 100212, 100230, 100302, 100320,
100410, 100500, 101004, 101022, 101040, 101112, 101130, 101202, 101220, 101310,
101400, 102003, 102012, 102021, 102030, 102102, 102111, 102120, 102201, 102210,
102300, 103002, 103020, 103110, 103200, 104010, 104100, 105000, 110004, 110022,
110040, 110112, 110130, 110202, 110220, 110310, 110400, 111003, 111012, 111021,
111030, 111102, 111111, 111120, 111201, 111210, 111300, 112002, 112020, 112110,
112200, 113010, 113100, 114000, 120003, 120012, 120021, 120030, 120102, 120111,
120120, 120201, 120210, 120300, 121002, 121020, 121110, 121200, 122010, 122100,
123000, 130002, 130020, 130110, 130200, 131010, 131100, 132000, 140010, 140100,
141000, 150000, 200004, 200022, 200040, 200112, 200130, 200202, 200220, 200310,
200400, 201003, 201012, 201021, 201030, 201102, 201111, 201120, 201201, 201210,
201300, 202002, 202020, 202110, 202200, 203010, 203100, 204000, 210003, 210012,
210021, 210030, 210102, 210111, 210120, 210201, 210210, 210300, 211002, 211020,
211110, 211200, 212010, 212100, 213000, 220002, 220020, 220110, 220200, 221010,
221100, 222000, 230010, 230100, 231000, 240000, 300003, 300012, 300021, 300030,
300102, 300111, 300120, 300201, 300210, 300300, 301002, 301020, 301110, 301200,
302010, 302100, 303000, 310002, 310020, 310110, 310200, 311010, 311100, 312000,
320010, 320100, 321000, 330000, 400002, 400020, 400110, 400200, 401010, 401100,
402000, 410010, 410100, 411000, 420000, 500010, 500100, 501000, 510000, 600000,
1000006, 1000042, 1000105, 1000132, 1000231, 1000321, 1000411, 1000420, 1000510,
1001014, 1001041, 1001050, 1001113, 1001140, 1001203, 1001302, 1001320, 1002022,
1002121, 1002202, 1002211, 1002301, 1002310, 1002400, 1003003, 1003030, 1003111,
1003300, 1004011, 1004101, 1004200, 1010023, 1010122, 1010212, 1010311, 1010401,
1010500, 1011004, 1011031, 1011121, 1011130, 1011211, 1011220, 1012012, 1012102,
1012120, 1012201, 1013011, 1013020, 1013110, 1015000, 1020013, 1020022, 1020040,
1020103, 1020202, 1020211, 1020220, 1020400, 1021003, 1021021, 1021030, 1021102,
1021111, 1021201, 1021210, 1021300, 1022101, 1023001, 1023100, 1030012, 1030021,
1030030, 1030111, 1030120, 1030201, 1031002, 1031101, 1032010, 1040002, 1040011,
1040101, 1040110, 1040200, 1042000, 1050001, 1100023, 1100050, 1100113, 1100122,
1100212, 1100302, 1100311, 1100401, 1100500, 1101004, 1101031, 1101121, 1101130,
1101211, 1101220, 1101310, 1102003, 1102012, 1102021, 1102102, 1102120, 1102201,
1102210, 1103002, 1103011, 1103020, 1103101, 1103110, 1103200, 1105000, 1110004,
1110013, 1110031, 1110040, 1110103, 1110112, 1110130, 1110202, 1110301, 1111012,
1111021, 1111030, 1111111, 1111120, 1111201, 1111210, 1111300, 1112002, 1112011,
1112020, 1113001, 1113010, 1113100, 1120021, 1120030, 1120102, 1120111, 1120120,
1120210, 1121002, 1121020, 1121101, 1122001, 1122010, 1122100, 1130011, 1130101,
1130110, 1130200, 1132000, 1140001, 1141000, 1200004, 1200013, 1200031, 1200121,
1200130, 1200202, 1200211, 1200220, 1201012, 1201021, 1201102, 1201120, 1201201,
1201210, 1202011, 1202020, 1202110, 1203100, 1204000, 1210003, 1210021, 1210102,
1210111, 1210201, 1210210, 1210300, 1212001, 1212100, 1220002, 1220101, 1221010,
1231000, 1300012, 1300030, 1300111, 1300201, 1300300, 1301011, 1301020, 1302001,
1310002, 1310020, 1311001, 1311010, 1311100, 1330000, 1401001, 1401100, 1410010,
1420000, 1500001, 1500010, 1500100, 2000005, 2000014, 2000131, 2000140, 2000203,
2000221, 2000311, 2000320, 2001013, 2001022, 2001202, 2001211, 2001400, 2002021,
2002030, 2002102, 2002111, 2002210, 2003020, 2003101, 2004001, 2004100, 2010004,
2010022, 2010031, 2010130, 2010202, 2010211, 2010220, 2010301, 2010400, 2011012,
2011030, 2011102, 2011111, 2011201, 2011300, 2012020, 2012101, 2012110, 2014000,
2020102, 2020111, 2020120, 2020300, 2021011, 2021110, 2022001, 2023000, 2030002,
2031001, 2031010, 2040010, 2100031, 2100112, 2100220, 2100301, 2101012, 2101021,
2101102, 2101111, 2101120, 2101201, 2102002, 2102110, 2103010, 2104000, 2110003,
2110021, 2110120, 2110210, 2111002, 2111011, 2111101, 2111200, 2120011, 2120020,
2130001, 2130100, 2200030, 2200111, 2200300, 2201101, 2210011, 2210110, 2212000,
2220001, 2300002, 2300011, 2300101, 2300200, 2401000, 3000004, 3000031, 3000112,
3000130, 3000301, 3001012, 3001030, 3001120, 3001201, 3002002, 3002011, 3002020,
3003010, 3010021, 3010102, 3010111, 3010210, 3011020, 3012001, 3012100, 3020101,
3031000, 3100003, 3100102, 3100111, 3100201, 3100300, 3102001, 3110002, 3111010,
3200020, 3201001, 3201100, 3220000, 3300010, 4000003, 4000021, 4000120, 4000210,
4001002, 4001011, 4001101, 4001200, 4010011, 4010020, 4020001, 4020100, 4100011,
4100110, 4102000, 4110001, 5000002, 5001010, 5110000, 6000001, 7000000]

P.S.  2110 is not a multiple of 4.

>>> Maximilian Hasler maximilian.hasler at gmail.com <<<<~!B*+R^&>>>> Wed Jan 6 04:13:25 CET 2010 <<<<~!B*+R^&>
The numbers with the property that
the constant sequence  (k,k,k,....) is a solution to the original problem
(without the requirement of being strictly increasing, of course),
are:
1, 11, 20, 102, 111, 120, 201, 210, 300, ...

This matches
A061384 : Numbers n such that sum of digits = number of digits.

but the next term in A061384, 1003, does not have any more the above property.

However, it is easy to see that the terms of A061384 which have the
additional property of being a multiple of their sum-of-digits,
(satisfied by all of the above terms except for 11),
do have the initially mentioned property ((k,k,k,...) is a solution).
That subsequence continues
1012, 1120, 1300, 2020, 2110, 2200, 3100, 4000, 10040, 10130, ...

The number 11 as well as 1111 lead to a (constant) solution in the
same way as does any repunit 11...11 = (10^m-1)/9.

Maximilian

On Tue, Jan 5, 2010 at 9:59 PM,  <hv at crypt.org> wrote:
> Assuming there is an additional rule that a(n+1) > a(n), I knocked up some
> (very hacky) code to search for longer sequences, attached below.
>
> Running it for half an hour, it gets as far as:
>  4 10 50 68 89 91 100 110 120 210 1000 2000 10000 20000 100000 100010 100080 100313 102000
> (and the initial seven terms were fixed after the first 10 seconds).
[snip]