Saturday, September 12, 2020

Casio fx-9750GIII: Happy Integers

Casio fx-9750GIII: Happy Integers


What Makes an Integer Happy?  

1.  Take a positive integer

2.  Take the sum of the squares of its integers

3.  Repeat the process until:


a.   You repeat a previous result or

b.   You obtain a sum of 1.  


In the cases that you eventually reach 1, that number is defined as a happy integer.


23:

2^2 + 3^2 = 13

1^2 + 3^2 = 10

1^2 + 0^2 = 1

23 is a happy integer.  


24:

2^2 + 2^2 = 4

4^2 = 16

1^2 + 6^2 = 37

3^2 + 7^2 = 58

5^2 + 8^2 = 89

8^2 + 9^2 = 145

1^2 + 4^2 + 5^2 = 42

4^2 + 2^2 = 20

2^2 + 0^2 = 4    (see first addition - repeat)

24 is not a happy integer.

The sequence of sums of squares of the integer's digits is known as a cherry sequence.  


Casio fx-9750GIII: HAPPY

(316 bytes)


"2020-08-20 EWS"

"N>0, INTEGER: N"? → N

{ 0 } → List 1

Intg log N → D

N ÷ 10^D → W

Lbl 3

0 → T

For 0 → I To D

T + (Intg W)^2 → T

(W - Intg W) × 10 → W

Next

Augment(List 1, { T }) → List 1

ClrText

Locate 1, 4, T

For 1 → I To 100

Next

T = 1 ⇒ Goto 1

For 1 → I To Dim List 1 - 1

T = List 1[ I ] ⇒ Goto 2

Next

T → W

Intg log W → D

W ÷ 10^D → W

Goto 3

Lbl 1

ClrText

Locate 1, 3, W

Locate 1, 4,"IS A HAPPY INTEGER"

Stop

Lbl 2

Locate 1, 3, N

Locate 1, 4, "NOT A HAPPY INTEGER"

Stop


Notes:

(1)  10^ is shown as a subscript 10 on the calculator.

(2)  The program sequence

ClrText

Locate 1, 4, T

For 1 → I To 100

Next

creates a timer.  This allows the calculator to show intermediate results for a short time.

(3)  Intermediate sums of squares are stored in List 1.  Other than the first 0 (an element is required to start a list in Casio programming), the rest of the sequence is known as Cheery Sequence.


Examples

N = 19;  Happy Number

N = 77; Not a Happy Number

N = 230; Happy Number

N = 562; Not a Happy Number 


Sources:

Duncan, Donald C.  Happy Integers.  The Mathematics Teacher, Vol 65. No. 7  November 1972, pp. 627-629  https://www.jstor.org/stable/27959021 

Happy Number.  GeeksforGreeks.  https://www.geeksforgreeks.org/happy-number  Updated February 4, 2020.  Accessed August 19, 2020.  (website under maintenance)

Eddie

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