Download View PDF - Journal of Computer and Mathematical Sciences

Journal of Computer and Mathematical Sciences, Vol.7(11), 613-619, November 2016
(An International Research Journal), www.compmath-journal.org
ISSN 0976-5727 (Print)
ISSN 2319-8133 (Online)
Non-successive Occurrence of a Non-Zero Digit in All Base
b Natural Numbers Less Than bn
Neeraj Anant Pande
Associate Professor,
Department of Mathematics and Statistics,
Yeshwant Mahavidyalaya (College), Nanded – 431602, Maharashtra, INDIA.
(Received on: November 16, 2016)
ABSTRACT
For a number system with any base b > 1, all natural numbers less than bn
for every positive integer n are considered. Analysis of non-successive occurrence of
the first natural number and non-zero digit 1 in first base b numbers smaller than bn is
done in this work. The formulae for count of the number of non-successive
occurrences of 1’s, their first and last instances are derived. All the analysis is
extended to multiple number of non-successive occurrences of 1’s. All results get
easily generalized for non-successive occurrences of all non-zero digits. As a
particular example, base b = 16 is illustrated with examples.
Mathematics Subject Classification 2010 : 11Y35, 11Y60, 11Y99.
Keywords: Natural numbers, non-zero digits, non-successive occurrence, base b.
1. INTRODUCTION
All counting numbers 1, 2, 3, ⋯ , beginning with the very first one and having at most
a fixed number of significant digits, are considered for search of non-successive non-zero
digits.
Decimal system in regular use has base 10 and 10 digits, viz., 0, 1, 2, 3, 4, 5, 6, 7, 8,
9. Any number system with base b > 1 is equally competent and has exactly b number of digits.
For b ≤ 10, digit notations are 0, 1, 2, ⋯ , b – 1 while for b > 10, first 10 digits are symbols 0,
1, 2, 3, 4, 5, 6, 7, 8, 9 and remaining are alphabets A, B, C, ⋯ , X, where X, not necessarily
actual alphabet X, is an alphabet at number equal to 9 more than its own count in the list of
alphabets.
For brevity, we will use term number here to denote a natural number. For each n ∈ N,
we consider the ranges 1 – bn, with bn left out. In such a range, the numbers are m, with
1 ≤ m < bn. If we use base b for these numbers, then all of them have digits less than or equal
to n.
613
Neeraj Anant Pande, Comp. & Math. Sci. Vol.7 (11), 613-619 (2016)
2. NON-SUCCESSIVE OCCURRENCE OF DIGIT 1
In present work, the non-successive occurrence of digit 1 is analyzed in the range of
1 – bn, except the last number bn, for all natural numbers n in base b.
All occurrences and successive occurrences of 1’s in base b numbers less than bn are already
formulated 8,9.
Theorem 1 : If r, n and b are positive integers with r ≤ n and b > 1, then the number of numbers
in base b containing exactly r number of digit 1’s in the range 1 ≤ m < bn is
A n
n
nr
.
1 Or  Cr (b  1)
b
A n
where the notation 1 Or is for count of base b numbers less than bn with r number of 1’s.
b
Theorem 2 : If r, n and b are positive integers with r ≤ n and b > 1, then the number of numbers
in base b containing exactly r number of successive digit 1’s in the range 1 ≤ m < bn is
S n
n( r 1)
C1 (b  1)nr .
1 Or 
b
S n
where the notation 1 Or is for count of base b numbers less than bn with r number of successive
b
1’s.
Choosing hexadecimal number’s base b = 16, we have determined these counts of nonsuccessive occurrences of single 1 and also double 1’s in numbers less than 1615, which is little
more than one quintillion (1018), by using a Java program. Although the base of numbers under
analysis is 16, we chose decimal number system writing their count!
Table 1 : Number of Hexadecimal Natural Numbers in Various Ranges with Single and Double
Non-successive 1’s in Their Digits
Sr.
No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Numbers Range Less Than
Number of Numbers in Base 16 with
single (Non-successive) 1
161
162
163
164
165
166
167
168
169
1610
1611
1612
1613
1614
1615
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
614
Number of Numbers in Base 16 with two Nonsuccessive 1’s
0
0
15
675
20,250
506,250
11,390,625
239,203,125
4,784,062,500
92,264,062,500
1,729,951,171,875
31,715,771,484,375
570,883,886,718,750
10,120,214,355,468,750
177,103,751,220,703,125
Neeraj Anant Pande, Comp. & Math. Sci. Vol.7 (11), 613-619 (2016)
In the first range 1 ≤ m < 161 = 16, single 1 comes only once which is not
non-successive.
In the second range 1 ≤ m < 162 = 256, single 1 comes total 30 times none of which,
being single, is non-successive.
Within this range, double 1 occurs once in number 11. But this is also not of
non-successive type.
In the third range, 1 ≤ m < 163 = 4,096, there is no non-successive single 1, but nonsuccessive double 1’s occur in 15 numbers
101, 121, 131,⋯, 1F1
at unit’s and hundred’s places.
All counts in above table come this way.
There is a formulation possible for the count of base b numbers with r non-successive
1’s in them in such ranges from 1 to 1 less than various base powers.
N n
Notation: We introduce the generalized notation 1 Or for count of base b numbers less than
b
b with non-successive r number of 1’s.
n
Theorem 3 : If r > 1, n > 2 and b are positive integers with 1 < r < n and b > 1, then the number
of base b numbers containing exactly r number of non-successive digit 1’s in the range
1 ≤ m < bn is
N n
n
n( r 1)
C1 )  (b  1)nr .
1 Or  ( Cr 
b
Proof. Let n > 2, 1 < r < n and b > 1 be positive integers. The reason for taking n > 2 is that
there cannot occur any number of non-successive 1’s in numbers < b2 and for r > 1 is that there
is no non-successive single 1 anywhere!
By Theorem 1, the number of base b numbers with all kinds of r number of 1’s in the range
1 ≤ m < bn is given by
A n
n
n r
.
1 Or  Cr (b  1)
b
By Theorem 2, the number of base b numbers with r number of successive 1’s in the range
1 ≤ m < bn is given by
S n
n( r 1)
C1 (b  1)nr .
1 Or 
b
Now the number of base b numbers with r number of non-successive 1’s in the range
1 ≤ m < bn will be difference of these two and hence,
N
1
Orn  A1 Orn  S1 Orn
b
b
b
 Cr (b  1) nr  n( r 1)C1 (b  1)nr
n
 ( nCr  n( r 1)C1 )  (b  1) nr
This completes the proof of the theorem.
615
Neeraj Anant Pande, Comp. & Math. Sci. Vol.7 (11), 613-619 (2016)
The table given above for hexadecimal numbers is now extended to higher occurrences
of non-successive 1’s.
Table 2: Number of Hexadecimal Natural Numbers in Various Ranges with Multiple
Non-successive 1’s in Their Digits
Sr. Number Range Number of Numbers in Base 16 Number of Numbers in Base Number of Numbers in
No.
<
with 3 Non-successive 1’s
16 with 4 Non-successive 1’s
Base 16 with 5
Non-successive 1’s
1.
164
30
0
0
2.
165
1,575
45
0
3.
166
54,000
2,700
60
4.
167
1,518,750
104,625
4,050
5.
168
37,968,750
3,290,625
175,500
6.
169
877,078,125
91,125,000
6,125,625
7.
1610
19,136,250,000
2,312,296,875
186,806,250
8.
1611
399,810,937,500
55,016,718,750
5,182,734,375
9.
1612
8,073,105,468,750
1,245,564,843,750
133,953,750,000
10.
1613
158,578,857,421,875
27,102,568,359,375
3,275,374,218,750
11.
1614
3,044,714,062,500,000
570,883,886,718,750
76,579,171,875,000
12.
1615
57,347,881,347,656,250
11,703,119,677,734,375 1,725,337,968,750,000
Table 2 : Continued …
Sr. Number Range Number of Numbers in Base Number of Numbers in Base
No.
<
16 with 6 Non-successive 1’s 16 with 7 Non-successive 1’s
1.
167
75
0
Number of Numbers in
Base 16 with 8
Non-successive 1’s
0
2.
168
5,625
90
0
3.
169
270,000
7,425
105
4.
1610
10,378,125
391,500
9,450
5.
1611
346,275,000
16,453,125
543,375
6.
1612
10,445,203,125
596,868,750
24,806,250
7.
1613
291,827,812,500
19,466,578,125
972,759,375
8.
1614
7,673,294,531,250
585,022,500,000
34,126,312,500
9.
1615
192,024,580,078,125
16,469,135,156,250
1,098,113,203,125
Sr.
No.
1.
2.
3.
4.
5.
6.
Number
Range <
1610
1611
1612
1613
1614
1615
Table 2 : Continued …
Number of Numbers in
Number of Numbers in Base
Base 16 with
16 with 10 Non-successive
9 Non-successive 1’s
1’s
120
0
11,700
135
729,000
14,175
35,943,750
951,750
1,515,712,500
50,422,500
56,930,343,750
2,275,846,875
616
Number of Numbers in
Base 16 with
11 Non-successive 1’s
0
0
150
16,875
1,215,000
68,850,000
Neeraj Anant Pande, Comp. & Math. Sci. Vol.7 (11), 613-619 (2016)
Sr. No.
1.
2.
3.
Number
Range <
1613
1614
1615
Table 2 : Continued …
Number of Numbers in Number of Numbers Number of Numbers in
Base
16
with in Base 16 with Base
16
with
12 Non-successive 1’s
13 Non-successive 1’s 14 Non-successive 1’s
165
0
0
19,800
180
0
1,522,125
22,950
195
3. FIRST NON-SUCCESSIVE OCCURRENCE OF DIGIT 1
Since single digit in any number cannot be considered as non-successive, the first
number containing 1 is not 1, in fact, by that convention there is no single non-successive 1 in
any number. For 2 non-successive 1’s, the first number containing it is 101, for 3 it is 1011
and so on. These representations are in corresponding base b systems. We can formulate it
easily.
Formula 1 : If n, r and b > 1 are natural numbers, then the first occurrence of r number of nonsuccessive 1’s in numbers in base b in range 1 ≤ m < bn is

, if r  n

 r
f= 
 1 b j  , if r  n .
 j 0
 j r 1
4. LAST NON-SUCCESSIVE OCCURRENCE OF DIGIT 1
The last non-successive occurrences of 1 in hexadecimal system in ranges till powers
of 16 are as follows.
Table 3 : Last Hexadecimal Numbers with Multiple Non-successive 1’s in their Digits in Various
Base Power Ranges
Sr. Last Number with NonNo.
successive
1.
11
2.
2 1’s
3.
3 1’s
4.
4 1’s
5.
5 1’s
6.
6 1’s
7.
7 1’s
8.
8 1’s
161
-
162
-
163
1F1
-
164
F,1F1
1,F11
-
They come following a rule.
617
Number Range <
165
166
167
168
169
FF,1F1 FFF,1F1 F,FFF,1F1 FF,FFF,1F1FFF,FFF,1F1
F1,F11 FF1,F11 F,FF1,F11 FF,FF1,F11FFF,FF1,F11
1F,111 F1F,111 F,F1F,111 FF,F1F,111 FFF,F1F,111
1F1,111 F,1F1,111 FF,1F1,111 FFF,1F1,111
1,F11,111 F1,F11,111 FF1,F11,111
1F,111,111 F1F,111,111
1F1,111,111
Neeraj Anant Pande, Comp. & Math. Sci. Vol.7 (11), 613-619 (2016)
Formula 2 : If n, r and b > 1 are natural numbers, then the last occurrence of r non-successive
1’s in numbers in base b in range 1 ≤ m < bn is

, if r  n


r
n

1

l= 
1 b j     (b  1)  b j , if r  n .


 j 0
j r 1

j r
 j r 1
5. EXTENSION TO OTHER NON-ZERO DIGITS
All formulae for occurrences – regarding count, first and last one – of non-successive
digit 1’s are extendable to all other non-zero digits. Denoting any non-zero digit by d, 1 ≤ d < b,
we have following.
N n
Notation : We further generalize the notation d Or for number of base b numbers less than bn
b
with r number of non-successive digits d’s.
Theorem 4 : If r, n, d and b are positive integers with 1 < r < n and 1 ≤ d < b, then the number
of base b numbers containing exactly r number of non-successive digit d’s in the range
1 ≤ m < bn is
N n
n
n  ( r 1)
C1 )  (b  1)n r .
d Or  ( Cr 
b
Formula 3 : If n, r, d and b are natural numbers with 1 ≤ d < b, then the first occurrence of r
number of non-successive digit d’s in base b numbers in range 1 ≤ m < bn is

, if r  1 or r  n


 r
f= 
d  b j  , if r  1 and r  n .


 j 0

 j  r 1
Formula 4 : If n, r, d and b are natural numbers with 1 ≤ d < b, then the last occurrence of r
number of non-successive digit d’s in base b numbers in range 1 ≤ m < bn is

, if r  1 or r  n


n 1
 r
l= 
d  b j     (b  1)  b j , if r  1 and r  n .


 j 0
j  r 1

j r
 j  r 1
Various integer sequences that come out of all formulae here are much unexplored yet.
Remark : All the work here has generalized some earlier results4.
618
Neeraj Anant Pande, Comp. & Math. Sci. Vol.7 (11), 613-619 (2016)
6. ACKNOWLEDGEMENTS
The author is thankful to Development Teams of Java Programming Language,
NetBeans IDE, and Microsoft Office Excel. These software were immensely useful in
performing rigorous checks.
The author is thankful to the anonymous referees of this paper.
REFERENCES
1. Neeraj Anant Pande, “Numeral Systems of Great Ancient Human Civilizations”, Journal
of Science and Arts, Year 10, No. 2 (13), pp. 209-222 (2010).
2. Neeraj Anant Pande, “Analysis of Occurrence of Digit 1 in Natural Numbers Less Than
10n”, Advances in Theoretical and Applied Mathematics, 11(2), pp. 99-104 (2016).
3. Neeraj Anant Pande, “Analysis of Successive Occurrence of Digit 1 in Natural Numbers
Less Than 10n”, American International Journal of Research in Science, Technology,
Engineering and Mathematics, 16(1), pp. 37-41 (2016).
4. Neeraj Anant Pande, “Analysis of Non-successive Occurrence of Digit 1 in Natural
Numbers Less Than 10n”, International Journal of Advances in Mathematics and
Statistics, Accepted, (2016).
5. Neeraj Anant Pande, “Analysis of Occurrence of Digit 0 in Natural Numbers Less Than
10n”, American International Journal of Research in Formal, Applied and Natural
Sciences, Accepted, (2016).
6. Neeraj Anant Pande, “Analysis of Successive Occurrence of Digit 0 in Natural Numbers
Less Than 10n”, IOSR-Journal of Mathematics, Vol.12, Issue 5, Ver. VIII, pp 70 – 74
(2016).
7. Neeraj Anant Pande, “Analysis of Non-successive Occurrence of Digit 0 in Natural
Numbers Less Than 10n”, International Journal of Emerging Technologies in
Computational and Applied Sciences, Accepted, (2016).
8. Neeraj Anant Pande, “Analysis of Occurrence of a Non-Zero Digit in All Base b Natural
Numbers Less Than bn”, International Journal of Computational Science and
Mathematics, Communicated, (2016).
9. Neeraj Anant Pande, “Successive Occurrence of a Non-Zero Digit in All Base b Natural
Numbers Less Than bn”, International Journal of Mathematics Trends and Technology,
Accepted, (2016).
10. Nishit K. Sinha, “Demystifying Number System”, Pearson Education, New Delhi, (2010).
619