**Playing with digital roots**

(*DR*, in short)

“The digital root
(also *repeated digital sum*) of a
number is the number obtained by adding all the digits, then adding the digits of
that number, and then continuing until a single-digit number is reached. For
example, the digital root of 65536 is 7, because 6 + 5 + 5 + 3 + 6 = 25
and 2 + 5 = 7.”

Now, what would be
the digital root of DIGITAL ROOT? I’ve used Gef’s Gematron
(a=1, b=2, c=3,... z=26) and found 4. And this 4 is
the same figure as the DR of *DR* (and
also of *dumb*, *lame*, *imbecile*, I know!)

What about the DR
of a few (English) number names? (Note that the DR of a sum of elements is the
same as the DR of the concatenation of the said elements):

ONE [7]

TWO [4]

THREE [2]

FOUR [6]

FIVE [6]

SIX [7]

SEVEN [2]

NINE [6]

TEN [3]

ELEVEN [9]

TWELVE [6]

THIRTEEN [9]

FOURTEEN [5]

FIFTEEN [2]

SIXTEEN [6]

SEVENTEEN [1]

EIGHTEEN [1]

NINETEEN [5]

TWENTY [8]

...

We see that no
integer is its own digital root (in English); what about a sequence of number
names divisible by its own DR? [This is not A064807, of
course (“Numbers which are divisible by their digital root”), we deal here with
*words*]. The first integers matching
the requisite are TWELVE [6], then SEVENTEEN [1], etc.:

S_{DR} =
12,17,18,23,25,27,30,...

__________

More seriously (?)
now; the few (complicated? artificial?) sequences hereunder refer somehow to
themselves.

T_{DR}, the first one,
is monotonically increasing: “a(n) and the sum
[a(1)+a(2)+a(3)+... a(a(n))] share the same DR”.

T_{DR} = 1, 3, 8, 9, 10,
11, 12, 17, 19, 28, 37, 46, 47, 48, 49, 50, 54, 55, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74,...

T_{DR} was build with the help of
this array, where **n** goes from 1 to
infinity; T_{DR}
is the sequence; **Q** is the cumulative
sum of T_{DR}’s
terms; **DR** is the digital root of **Q** (the underlined terms of **Q** are also the successive digital roots
of T_{DR} *and* the digital roots of
the 3 terms just above them):

**n**_{DR}=__1__|2|
__3__| 4| 5| 6| 7| __8__| __9__| __10__| __11__| __12__| 13|
14| 15| 16| __17__| 18| __19__| 20| 21| 22| 23| 24| 25| 26|
27| __28__| 29|...

T_{DR}=__1|3| 8| 9|10|11|12|17|19| 28| 37|
46| 47| 48| 49| 50| 54| 55| 64| 65| 66| 67| 68| 69| 70| 71|
72| 73| 74__|...

**Q**_{DR}=__1|4|12|21|31|42|54|71|90|118|155|201|248|296|345|395|449|504|568|633|699|766|834|903|973|1044|1116|1189|1263__|...

**DR**_{DR}=__1__|4| __3__| 3| 4|
6| 9| __8__| __9__| __1__| __2__|
__3__| 5| 8| 3| 8| __8__| 9| __1__| 3|
6| 1| 6|
3| 1| 9|
9| __1__| 3|...

In building T_{DR}, the smallest integer not present
and not leading to a contradiction was used. The sequence can be read like
this:

- Pick up any term of T_{DR}, for instance 12; now the digital root
of 12 (which is 3) is the same as the digital root of the sum of the first 12
terms of the sequence (which is 201, with digital root —>3)

- Another example is 19: the DR of 19 (—>1) is the same as
the DR of the sum of the first 19 terms of T_{DR} (568 —>1).

-----

U_{DR} is the same
sequence, dropping the ‘increasing’ constraint (we thus keep this definition
for U_{DR}:
: “a(n) and the sum [a(1)+a(2)+a(3)+... a(a(n))] share the same DR”.

U_{DR} = 1, 3, 8, 5, 6,
10, 9, 11, 19, 28, 37, 13, 16, 15, 23, 46, 18, 20, 21, 26, 55, 24, 32, 64, 27,
29, ...

U_{DR} was build with the same kind
of array:

**n**_{DR}=__1__|2|
__3__| 4| __5__| __6__| 7| __8__| __9__| __10__| __11__|
12| __13__| 14| __15__| __16__| 17| __18__| __19__| __20__| __21__|
22| __23__| __24__| 25| __26__| __27__| __28__| __29__|...

U_{DR}=__1|3| 8| 5| 6|10| 9|11|19| 28| 37|
13| 16| 15| 23| 46| 18| 20| 21| 26| 55| 24| 32| 64| 27| 29| 73| 82| 91__|...

**Q**_{DR}=__1|4|12|17|23|33|42|53|72|100|137|150|166|181|204|250|268|288|309|335|390|414|446|510|537|566|639|721|812__|...

**DR**_{DR}=1|4| 3| 8| 5| 6| 6| 8|
9| 1|
2| 6| 4| 1| 6|
7| 7| 9|
3| 2| 3|
9| 5| 6|
6| 8| 9|
1| 2|...

U_{DR} is the sequence showing the
lowest quantity of missing integers (those are the *not* underlined **n**’s above: 2,4,7,12,...)

To understand this sequence, just read it like
this:

- Select any term of U_{DR}, for instance 11; now the digital root
of 11 (—>2) is equal to the digital root of the sum of the first 11 terms of
U_{DR}
(which is 137 —> 1+3+7=11 —> 1+1 —> 2).

- Try 28; the digital root of 28 (—>1) is the digital root of the
sum of the first 28 terms (721—>1); etc.

In building the sequence from scratch, always
use the smallest integer not used so far and not leading to a contradiction.

_________

The same idea is at work hereunder, but involves
**prime numbers**:

V_{DR} is a re-ordering of the
Primes where a(n) and a(a(n)) share the same DR:

V_{DR} = 2, 11, 3, 5,
23, 7, 43, 13, 17, 19, 29, 41, 31, 61, 67, 71, 53, 73, 37, 83, 97, 107, 113,
127, ...

V_{DR} was build with a simple
two-line array where an (underlined) Prime in **n** has the same DR as the term below it, term which was the smallest
unused so far in building the sequence:

**n**_{DR}=1| __2__|
__3__| 4| __5__| 6| __7__| 8| 9|10|__11__|12|__13__|14|15|16|__17__|18|__19__|20|21|
22| __23__| 24| 25| 26| 27| 28|__29__| 30| __31__| 32| 33|...

V_{DR}=2|11| 3| 5|23| 7|43|13|17|19|29|41|31|61|67|71|53|73|37|83|97|107|113|127|137|139|149|181|47|191|193|197|223|...

V_{DR} is the lexicographically
first such sequence and could be read like this:

- Choose any term in V_{DR}, for
instance 11; now
the digital root of 11 (—>2) is the same as the digital root of the 11^{th}
term of the sequence (which is 29, with DR —>2).

- A try with 23; the DR of 23 is 5 and
the DR of the 23^{rd} term (113) is 5 too.

-----

Breaking News (March 3^{rd}, 2010):

**Georges Brougnard** has just corrected V_{DR} and sent a
nice graph of the first 500 terms:

Here are the 500 first terms of V_{DR} computed
by **Georges**:

V_{DR} = 2, 11, 3, 5, 23, 7, 43, 13, 17, 19, 29, 41, 31, 61, 67, 71, 53, 73, 37, 83,
97, 107, 113, 127, 137, 139, 149, 181, 47, 191, 193, 197, 223, 229, 239, 251,
263, 271, 277, 293, 59, 317, 331, 337, 347, 359, 367, 379, 389, 401, 409, 433,
439, 449, 461, 491, 499, 503, 523, 547, 79, 557, 563, 569, 577, 607, 103, 617,
619, 631, 89, 641, 109, 647, 661, 677, 691, 701, 727, 743, 751, 757, 101, 773,
787, 797, 827, 829, 839, 859, 881, 907, 911, 919, 941, 947, 151, 967, 971, 991,
1013, 1019, 1033, 1039, 1049, 1063, 179, 1069, 1097, 1109, 1123, 1129, 131,
1153, 1181, 1187, 1193, 1231, 1237, 1259, 1279, 1283, 1289, 1303, 1319, 1327,
163, 1361, 1367, 1399, 1409, 1423, 1429, 1451, 1471, 1481, 173, 1483, 157,
1493, 1523, 1531, 1543, 1553, 1567, 1583, 1597, 1607, 167, 1613, 1619, 1627,
1657, 1699, 1709, 1721, 1723, 1733, 1747, 1759, 1789, 1801, 1823, 1831, 1847,
1879, 1889, 1913, 1949, 1951, 1973, 1979, 1993, 1999, 2029, 2039, 2053, 2063,
2069, 2099, 199, 2113, 2129, 2137, 2141, 2143, 2179, 2203, 2207, 2239, 227,
2267, 211, 2269, 2273, 2297, 233, 2311, 2341, 2351, 2357, 2371, 2381, 2383,
2411, 2417, 2441, 2467, 2473, 2477, 2521, 2549, 2551, 2557, 2579, 2591, 2647,
2657, 2663, 2671, 2677, 2687, 241, 2707, 2711, 2719, 2731, 2753, 283, 2767, 2777,
2801, 2833, 2837, 2843, 2857, 2879, 2887, 257, 2917, 2927, 2939, 2969, 2971,
2999, 3011, 3019, 3041, 3067, 3079, 269, 3109, 3119, 3181, 3187, 3191, 3203,
3229, 3251, 3253, 3257, 3299, 281, 3307, 3313, 3319, 3347, 3359, 3361, 3389, 307,
3391, 3449, 3457, 3461, 3469, 313, 3499, 3527, 3529, 3539, 3559, 3571, 3583,
3593, 3617, 3623, 3631, 3643, 3673, 3677, 3709, 311, 3719, 3733, 3761, 3793,
3797, 3803, 3823, 3851, 3853, 3881, 3907, 3911, 3919, 3931, 3947, 3989, 4003, 4013,
4019, 4051, 4057, 4079, 4091, 353, 4099, 4111, 4139, 4177, 4201, 4211, 4217,
4241, 4243, 4271, 4273, 4283, 4297, 349, 4339, 4349, 4363, 4373, 4397, 373,
4409, 4441, 4447, 4457, 4481, 4519, 4523, 4547, 4549, 383, 4561, 4603, 4637, 4639,
4643, 4651, 4663, 4691, 4703, 4729, 4733, 431, 4759, 4787, 4793, 4801, 4817,
4831, 4861, 421, 4889, 4931, 4937, 4951, 4957, 4987, 4999, 5003, 5009, 5011,
5039, 397, 5051, 5059, 5099, 5113, 5119, 5171, 5179, 5227, 5231, 443, 5233, 5237,
5261, 5297, 5309, 5347, 5351, 5393, 5399, 5407, 5413, 419, 5437, 5441, 5449,
5471, 5479, 5501, 5521, 463, 5531, 5563, 5573, 5623, 5647, 5651, 5653, 5657,
5659, 5701, 5711, 5717, 5741, 5743, 5791, 5801, 5821, 5843, 5849, 5851, 5857, 5879,
5881, 487, 5903, 5923, 5981, 5987, 6007, 457, 6011, 6047, 6073, 6079, 6089,
6121, 6131, 6151, 6163, 467, 6173, 6197, 6217, 6221, 6229, 6287, 6299, 6301,
6311, 6343, 6353, 479, 6359, 6367, 6373, 6379, 6451, 6473, 6481, 6521, 6529, 6551,
6563, 6571, 6577, 6599, 6619, 6659, 6661, 6673, 6689, 6709, 6719, 6733, 6779,
6781, 6791, 6793, 6841, 6857, 6863, 509, 6869, 6883, 6907, 6917, 6947, 6967,
6977, 571, 6997,...

If someone needs more terms (for instance 666)
just copy/paste the hereunder blue script into Georges’s
**GBgraph** (which
can be found here), then
change manually **aDim****=100** into **aDim****=666**, then click “Draw” to plot
and “a(n)” to see the 666 integers wanted:

**aDim****=****100;**

**interval****(****1,aDim);**

**if(****inInit****)**

**{**

**aList**** = new Array(0);**

**function**** DR(n) {if(n <10) return n; return DR(digitsum(n));}**

**function**** nextDRprime(dr) {for (var p = notused(1);; p = notused(p))**

** if(isprime(p))**

** if(dr == -1 || DR(p) == dr) return p;**

** }**

** **

**for(****i=1;i<=aDim;i++)**

**if(****aList****[i] == undefined)**

** {**

** use(aList[i] = nextDRprime(-1))
;**

** if(aList[i] != i) use(aList[aList[i]] = nextDRprime(DR(aList[i])));**

** }**

**}**

-----

W_{DR} has a slightly different
definition from V_{DR}
– which makes all the difference: we allow now the presence of composite
numbers – whose quantity has been maximized here.

W_{DR} definition: a(a(n)) is a Prime sharing a(n)’s DR.

Again, a two-line array suffices:

**n**_{DR}=___1|2|3| 4|5| 6|7| 8|9|10|11|12|13|14|15|16|17|18|19|20|21|22|23|24|25|26|27| 28|29|30| 31|32|33|34|35| 36|__...

W_{DR}=19|2|3|13|5|1|7|17|4|37|11|
8|31|23|10|43|53|14|73|29|16|67|41|20|61|71|22|109|47|25|103|59|26|79|28|109|...

W_{DR} is the sequence showing the
lowest quantity of missing integers; those missing integers are the multiples
of 3 (except 3 itself), as the DR of a prime >3 is never 3, 6 or 9.

W_{DR} is the lexicographically
first such sequence and could be read like this:

- pick up any term in W_{DR}, for
instance the first one, 19;
now the digital root of 19 (—>1) is the same as the digital root of the 19^{th}
term of the sequence, which is a prime (indeed, the 19^{th} term is 73,
prime with DR —>1).

- a
try with 17; the DR
of 17 is 8 and the DR of the 17^{th} term (53, a Prime) is 8 too.

As usual, in building the sequence from scratch,
always use the smallest integer not present so far in W_{DR} and not leading to a
contradiction.

Best,

É.

_______________________

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