On Smarandache sequences and subsequences

Smarandache

On

Tl3Ilg Zhengping (Hanglhou Teacher's College,

Abstract

A

sequences and subsequences

Xu Kanghua Fuyang No.2 Middle Schoo~ Zhejiang, P. R. China)

Smarandache

sequence

partial

perfect

additive

sequence is studied completely in the first paragraph. In the second paragraph both Smarandache square-digital subsequence and squarepartial-digital subsequence are studied

Key

words

Smarandache subsequence.

ยง1 The be

a

Smarandache

square-digital

partial

perfect

subsequence, Smarandache

Smarandache partial

additive

square-partial-digital

perfect additive

Smarandache partial perfect additive

sequence

sequence,

is

sequence defined

to

sequence: 1, 1, 0, 2, -1, 1, 1, 3, - 2, 0, 0, 2, 0, 2, 2, 4,

-3,-1, -1, 1, -1, 1, 1, 3, -1, 1, ....

sequence has

This

P

j

i=1

and

p~1.

constructed in the fonowing way:

In [1]

function

for all

j=p+l

a 1p+1

(a)

that:

property

lp

La = La), It is

the

= ap +1 + 1

for all

M. Bencze raised

Can

you,

readers,

p

~ 1.

the following fmd

a

of n )?

146

two

general

questions: expression

of

an (as

Is it periodical or convergent or boWlded? (b) Please design (invent) yourselves other Smarandache (or partial perfect) f-sequences.

In this paper we solved the question (a) completely.

Suppose which

among

numbers

&l;

of

makes

8j

Thus

binary

the

&j

notation

=1,&, =0

of

as

n(n 2 2)

or 1 (i=O,l,路路路,k-l).

=l(i =0,1, .. 路,k),

g(n)

is

n

perfect

=(8A;8A;_1 "'8 8

Define

1

fen)

the

minimum

of

a(n) (i.e. a,,)

of

0

)2.'

are

the

i

that

as

the

=1. we

may

prove

the

expression

following:

r

1-

a(n) =

k, if k + 2f(n) + 2g(n) - 3 otherwise

We may

80

= 8 1 = '"

= 8):_1 = 0,

use mathematical induction to prove it

a(2) = 1,a(3) = 0= -1+ 2 x2+ 2 xO- 3= -1+ 2f(3) + 2g(3) -3.

So

the conclusion is valid for

Suppose

that

2,3"", n -len 2 3) .Let's

the

n =2,3,

conclusion

consider the

is

cases of

also

valid

for

n.

=k-l+l=k.

2

When

cases

should

not be

all

the

8 0,81> "',8):_1

discussed.

147

are

zeroes,

two

kinds

of

(1) If

Co

= o.

n

g(n) = g«EJc Elt:-l "'E1 EO )2) = g«E/cElt:-l "'E1 )2)+ 1 = g(i) + 1.

According

inductive hypothesis, we have n n n a(n) = a(-) + 1 =-(k-l)+ 2f(-) + 2g(-)- 3+ 1 to

2

2

2

=-k+ 2f(n) + 2(g(n) -1)-1 =-k+ 2f(n)+2g(n)-3.

(2) If

= 1,

Co

three

subcases

exist

the

notation

[xl

(i) If E1 =0.

=f{~J+ 1), more

than

denotes

the

it's

integer

x.

g(n) = g«Et Et _1'''E1Eo)2) = g«&t&t-l '''E3&2 1)2) =

so,

greatest

easily

a(n) =

known from

g{~J+ 1) = o.

inductive hypothesis

aC[~J+ 1)-1 =-(k-1)+ 2f{~]+ 1)+ 2gC[~J+ 1) -3-1

=-k+ 2f(n)+ 2g(n)-3.

148

not

=(EkEk_1 "'Ei+ll~l' i

So,

tiIfIa

/{~J+l)=/(n)-i, g{~]+l)=i=i+g(n).

According to

Then,

a(n) =

inductive hypothesis.

we

have

a<[~J+ 1)-1 =-(k-l)+ 2f{~J+ 1)+ 2g<[~J+ 1)-3-1

= -k+ 2(f(n)- i)+ 20 + g(n禄 - 3

=-k+ 2j(n) + 2g(n)-3.

j(n)=k+l, g(n)=O.

= (l~2'

SO

[~]+I=(e.te.t_l'路'E2el)2 +(1)2

from

1

k Ilma

Then

a{~]+I)-1 = k-l

a(n) =

= -k+ 2(k+ 1)+ 2 x 0- 3 = -k+ 2/(n) + 2g(n) -3.

From the natural numbers

above,

conclusion is

the

(a)

for

all

the

n(n ~ 2).

Having proved above fact, question

true

can be solved

a(n) =k,

so sequence {a(n)}

periodical

and

easily. is

the

remaining problem in

For if

unbounded,

convergent.

149

n = 2,t,

we have

therefore

cannot be

§2

Smaranche square-digital subsequence and

Smaranche square-partial-digital subsequence The Smaranche subsequence:

square-digital

is

subsequence

defined

be

to

a

0, 1, 4, 9, 49, 100, 144, 400, 441,

terms and

all

are

digits

those

perfect

2 71 ,

we

choose

(Therefore

squares

only

the

0, 1,

only

4

9)

In [1] the

...,

0, 1, 4, 9, 16, 25, 36,

from

i.e.

form

MBencze

questioned:

where

NO···O

-.-'

Disregarding

is

N

also

a

the

square

perfect

square,

numbers how

of

many

2.t timu

other

numbers We

sequence

find by

In fact, the

belong

to

that

this

sequence?

1444, 11449,

calculating. we may

find

also

491401,

infinitely

many

numbers

that

to

the

belong

to

be

a

sequence.

(2·10 k +1)2

=

4·102.1: + 4·10 k + 1,

Smarandache sequence:

square-partial-digital

49, 100, 144, 169,

and

9 = 32, etc.).

In

the

same

nmnbers

of the

way

form

subsequence

~

1. is

other

it

be partitioned into groups squares (169 can be partitioned

18

questioned:

-.-' where

NO···O

nmnbers

to

can

N

is

also

belong

to

150

this

sequence?

a

of as

the

square

perfect

square,

Disregarding

2.t timG

many

de:fmed

400, 441, ...

i.e. the square nmnbers that digits which are also perfect 16 = 4 2

k

all

for

how

belong

We or

22

numbers

in

1he

-1

form

aOb (here

a

neither

zero).

is

b

find

may

10404, 11025, 11449, 11664, 40401, 41616, 42025, 43681, 93025, 93636, 161604, 164025, 166464, 251001, 254016, 259081, 363609, 491401, 641601, 646416, 813604, 819025.

We

may

middle

the

701 2 ,

construct

infinitely

numbers

by

adcting

zero

in

of there numbers like 102 2 , 105 2 , 107 2 , 108 2 , 201\ 204 2 ,

8012 , 804 2 , 902 2 , 905 2

numbers

many

as

the

as

well.

we

may

find

some

other

following:

32436Q!, 102464Q1 25664Q1, !Q36324,

~Q64256,

36144144, 49196196,

81324324, 6425625§, 121484484, 169676676, 196784784, 484484121, 576576144, 676676169, 784784196 900900225, 1442401, 32436Q1 4004001

§3

Smaranche

cube-partial-digital

subsequence

1000, 8000, 10648, 27000, ... i.e. digits

the which

cube nmnbers that can be partitioned into groups are also perfect cubes (10648 can be partitioned

1=1\0=0\64=4\ and

8=2 3 )

Same question: disregarding M 0···0 IS also ............... ,where M

---

of as

the cube a perfect

numbers of the cube, how many

fonn: other

3k timG

numbers belong to this sequence?

As the above said, we may find belong to the sequence as well

151

infmitely

many

numbers

that

for all for

example

21818127 =3033 ,

216648648216 =6006 3

,

21610801800! =6001 3 , !Q!8108216 =10063

â&#x20AC;˘

k~

0

REFERENCES 1. Bencze, M(I998). "Smarandache Relationships and Subsequences", Bulletin of Pure and Applied Sciences. Vol 17E (NO.1) 1998; P.55-62.

152