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
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
