Sequence of Integers Generated by Summing the Digits of their Squares

Objectives: To establish some properties of sequence of numbers generated by summing the digits of their respective squares. Methods/Analysis: Two distinct sequences were obtained, one is obtained from summing the digits of squared integers and the other is a sequence of numbers can never be obtained be obtained when integers are squared. Also some mathematical operations were applied to obtain some subsequences. The relationship between the sequences was established by using correlation, regression and analysis of variance. Findings: Multiples of 3 were found to have multiples of 9 even at higher powers when they are squared and their digits are summed up. Other forms are patternless, sequences notwithstanding. The additive, divisibility, multiplicative and uniqueness properties of the two sequences yielded some unique subsequences. The closed forms and the convergence of the ratio of the sequences were obtained. Strong positive correlation exists between the two sequences as they can be used to predict each other. Analysis of variance showed that the two sequences are from the same distribution. Conclusion/Improvement: The sequence generated by summing the digits of squared integers can be known as Covenant numbers. More research is needed to discover more properties of the sequences. Sequence of Integers Generated by Summing the Digits of their Squares H. I. Okagbue1*, M. O. Adamu2, S. A. Iyase1 and A. A. Opanuga1 1Department of Mathematical Sciences, Covenant University, Canaanland, Ota, Nigeria; hilary.okagbue@covenantuniversity.edu.ng 2Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria


...(A).
This sequence is called the square number (integers) which can be found on the online encyclopedia of integer sequence A000290 -OEIS.
Equation (B) is obtained from (A) and some few examples are as follows: 5 = 1 + 4, 14 = 5 + 9, 30=14+16 The square number is the square of number (in this case an integer) and is the outcome when an integer is multiplied with itself 1 .
Many authors have worked on the square number but this paper introduces a new concept/property of the square number (integers) by investigating and examines the phenomenon of summing up the digits of squared numbers.Weissten 2 enumerated some characteristics of the square number while some theoretical aspects can be found in 3 .Some other literatures about the square numbers are as follows: Consecutive integers with equal sum of squares 4 .
Mixed sum of Squares and Triangular Numbers [5][6][7][8] .The Sum of digits of some Sequence 9,10 .The sum of Digits function of Squares 11,12 .Reducing a set of subtracting squares 13 .Squares of primes 14 .Sequences of squares with constant second differences 15 .Relationship between sequences and polynomials 16 .Square free numbers 17 .The sum of squares and some sequences 18 .
Number sequences have been applied in real life in modeling, simulation and development of algorithms of some carefully studied phenomena 19,20 .
• It increases as the integers increases.
• The ratio of two square integers is also a square.
• A square number is also the sum of two consecutive triangular numbers 21,22 • A square number cannot be a perfect number 23 .

Methodology
The first 3000 integers are squared and their respective digits summed up.The first 10 numbers, their square and the sum of their respective digits are summarized in Table 1.

Findings
Since the first 3000 integers are used, it was observed that a sequence of numbers is obtained and can be grouped in two distinct ways.First, when an integer is squared, and the digits summed, the following numbers can be obtained at varying frequencies which form the following sequence;

The Patternless Nature of the Sequences of Odd and Even Integers when the Digits of their Squares are Summed
Table 2 shows the results when the numbers are divided into two distinct equivalence classes of the odd and even integers.
There is no significance pattern of sequence formed by each class except the multiples of 3. "Hence we state that an even integer when squared and its digits summed yields even or odd integer and the same applies to any odd integer".The first 40 numbers of both Fibonacci and Lucas • sequences were squared, the sum of their respective individual numbers were obtained and the results are represented in a component bar chart.

. (E).
Table 3 shows some integers multiples of 3, their square and their respective sum of digits: "Hence we state that any integer divisible by 3, if squared and its digits summed yields an integer divisible by 9".

Higher Powers of Multiples of 3
Even at higher powers of the multiples of 3, the same result is obtained as shown in Table 4.As seen from the chart, the Lucas numbers increases more rapidly than the Fibonacci numbers.

Subsequences of Sequence C
Each of the numbers of sequence C also forms a sequence.For example, the 100 natural numbers can be grouped based on the numbers in sequence C.As seen from the chart, the Lucas numbers increases more rapidly than t Fibonacci numbers.

Subsequences of Sequence C
Each of the numbers of sequence C also forms a sequence.For example, the first 100 natural numbers can be grouped based on the numbers in sequence C.

Additive Properties
• Addition of two numbers of sequence (C) can yield numbers in both sequences (C) and (D).• Addition of two numbers of sequence (C) can produce numbers in the same sequence if; (a) A multiple of 9 is added to any numbers ofsequence (C).(b) A multiple of 9 are added to each other.
• A pattern can be formed from the addition of the numbers of sequence which can be seen from Table 1.• Addition of two numbers of sequence (D) yield no pattern but a patterned triangle similar to Paschal can be obtained which contained some numbers of sequence (C) in unique arrangement.

Multiplicative Properties
• The multiplication of any two numbers of sequence (C) yield a number in the same sequence.
• The multiplication of any two numbers of sequence (D) does not necessarily yield a number in the sequence.• 3. Multiplication of two numbers of sequence (D) yield no pattern but a patterned triangle similar to Paschal can be obtained which contained some numbers of sequence (C) in unique arrangement.

Uniqueness of Sequences C and D
Sequences (C) and (D) are unique.The complete respective sequences cannot be obtained by increment or decrement of the numbers in the sequences rather various sequences is obtained.When 1 is added to all the numbers in sequence (C), we obtain;

Multiplicative Properties
The multiplication of any two numbers of sequence (C) yield a number in the same sequence.yield a number in the sequence.
3. Multiplication of two numbers of sequence (D) yield no patte patterned triangle similar to Paschal can be obtained which con numbers of sequence (C) in unique arrangement.

. (IE).
Here it can be seen that sequence (IE) is closely related to sequence (D).

The Ratio of Sequence (C)
The ratio of the two successive integers of sequence (C) is as follows: 4 1

The Ratio of Sequence (D)
The ratio of the two successive integers of sequence (C) is as follows: 3 2

The Sequences Obtained from the Various Factors of Sequences (C) and (D)
The first 40 members of sequences C and D are listed.Some subsequences are obtained by the various factors such as 2n, 3n, 4n...

Factors of 2
Subsequence is formed for both sequences C and D if they are arranged based on the factors of two.The first is

Regression Analysis of the First 40 Terms of Sequences C and D
Since there is a strong positive correlation between the two sequences, the predictive capability of the sequences with respect to each other is analyzed using the regression for the first 40 terms of both sequences.

Sequence C as the Dependent Variable
The results of regression analysis of the two sequences when sequence C is the dependent variable and sequence D as the independent variable are summarized as follows; The R, adjusted R square, R square and R square change have the same value of 0.999.The regression equation is; C = 0.201 + 0.999D (1).The result of the analysis of variance is summarized in Table 10.

Sequence D as the Dependent Variable
The results of regression analysis of the two sequences when sequence D is the dependent variable and sequence C as the independent variable are summarized as follows; The R, adjusted R square, R square and R square change have the same value of 0.999.The regression equation is; D = -0.123+ 0.004C (2).The result of the analysis of variance is summarized in Table 11.

Test of Equality of Means
The sequences have the same mean effect as summarized in Table 12.
for sequence C and the second is for sequence D.

The Square of Sequences C and D
New sequences are obtained from the square of sequences C and D.

The Ratio of the Sequences
The ratio of the two sequences also produced some sequences.

Linear Correlation between Sequences C and D
There is a strong positive correlation between the two sequences.Pearson correlation coefficient is 0.999, Spearman rho is 1.0 and Kendall's tau is 1.0.

Conclusion
The paper have described the properties of sum of the digits of square numbers and their associated sequences and multiples of 3 were found to be the only class of integers with unique pattern when their digits of their square are summed.The closed form of the ratios gave approximate ratios.More research is needed to produce more features and properties of the sequences.The authors proposed that sequence C be named COVENANT NUMBERS and be included in the online encyclopedia of integer sequences database.

Figure 1 .
Figure 1.Component bar chart of the first 40 Fibonacci and Lucas number.

Figure 2 .
Figure 2. The first 100 numbers and their digits sum grouped in sequence C.

1 •
Every 4 th number of the sequence (C) is a multiple of 9.• As expected all the square numbers are in sequence (C).• All three consecutive numbers of sequence (C) are coprime but not pairwise gcd , , All four consecutive numbers of sequence (C) are coprime but not pairwise gcd , , ,

Figure 3 .
Figure 3. Binomial table obtained from addition of terms in sequence D.
, , , , ... (J)The sequence converges to almost one with a mean of 1.11200275.The closed form solution of the ratio can be written as: j = + , , , , ... (K)The sequence converges to almost one with a mean of 1.101494025.The closed form solution of the ratio can be written as:

Figure 5 .
Figure 5.The ratio of sequence (C).x axis -terms in sequence C; y axis -the ratio of 2 consecutive terms of sequence C.

Figure 6 .
Figure 6.The ratio of sequence (D).x axis -terms in sequence D; y axis -the ratio of 2 consecutive terms of sequence D.
23 Square number has an odd number of positive divisors23.Square Divisors Number

Table 1 .
The first ten terms, their square and digits

Table 2 .
The sum of the digits for odd and even numbers

Table 3 .
Some integers multiples of 3

Table 4 .
Higher powers of multiple of 3 and their sums of digits.

Table 5 .
Addition of terms of sequence C.

Table 7 .
The 11 th to 20 th terms of sequences C and D

Table 6 .
The first 10 terms of sequences C and D

Table 8 .
The 21 st to 30 th terms of sequences C and D Sequence of Integers Generated by Summing the Digits of their Squares Indian Journal of Science and Technology Vol 8 (15) | July 2015 | www.indjst.org

Table 9 .
The 31 st to 40 th terms of sequences C and D