HP Prime: Integer Ratios and Integer Density
How Many Integers Fit the Criteria?
The program INTDENS calculates the ratio of integers to a range of integer that fit a criteria. The seven ratios that the program offers are:
1. The number of odd integers over the range of integers
2. The number of even integers over the range of integers
3. The number of integers equally divisible by N over the range of integers. The user sets the value of N.
4. The number of perfect squares over the range of integers. An integer is a perfect square when the fractional part of integer is zero. Examples are perfect squares are 1, 4, 9, 16, and 25.
5. The number of integers that begin with digit N over the range of integers. N is the digit 0-9.
6. The number of integers that are triangular numbers over the range of integers. a triangular number is an integer of the form (n * (n + 1)) / 2.
7. The number of integers that are relativity prime to N over the range of integers. An integer is relatively prime to N when the greatest common divisor between N and that integer is 1.
The program can be used for testing whether density exists with the criteria. The density, if it exists, is defined as:
limit
n → ∞ (number of integers from 0 to n-1 that fit a criteria) / (n - 1)
For example: the integer density of odd integers is 1/2, while the integer density of integers divisible by 6 is 1/6. Beware if the limit tends towards 0 as n gets large, such as the number of perfect squares or triangular numbers.
HP Prime Program: INTDENS
Code:
EXPORT INTDENS()
BEGIN
// integer density
// 2023-03-20 EWS
LOCAL A,B,C,H,I,N,lst,E;
I:=0;
// list of choices
lst:={"Odd Integers",
"Even Integers",
"Divisible by N",
"Perfect Squares",
"Begins With Digit N",
"Triangular Numbers",
"Rel. Prime to N"};
INPUT({A,B,{H,lst}},
"Integer Density",
{"LOW: ","HIGH: ","TYPE: "});
// odd
IF H==1 THEN
FOR I FROM A TO B DO
C:=when(I MOD 2==1,1,0)+C;
END;
END;
// even
IF H==2 THEN
FOR I FROM A TO B DO
C:=when(I MOD 2==0,1,0)+C;
END;
END;
// divisible by N
IF H==3 THEN
INPUT(N,"Divisible by N","N: ");
FOR I FROM A TO B DO
C:=when(I MOD N==1,1,0)+C;
END;
END;
// perfect squares
IF H==4 THEN
FOR I FROM A TO B DO
C:=when(FP(√I)==0,1,0)+C;
END;
END;
// begins with digit N
IF H==5 THEN
INPUT(N,"Begins with digit N","N: ");
FOR I FROM A TO B DO
E:=ALOG(IP(LOG(I)));
E:=IP(I/E);
C:=when(E==N,1,0)+C;
END;
END;
// triangular numbers
IF H==6 THEN
FOR I FROM A TO B DO
E:=(−1+√(1+8*I))/2;
C:=when(FP(E)==0,1,0)+C;
END;
END;
// relatively prime to N
IF H==7 THEN
INPUT(N,"Relatively Prime to N","N: ");
FOR I FROM A TO B DO
C:=when(gcd(I,N)==1,1,0)+C;
END;
END;
// results
N:=B-A+1;
MSGBOX("Count: "+STRING(C)+"\n
Range: "+STRING(N)+"\n
Ratio: "+STRING(C/N));
RETURN QPI(C/N);
END;
Examples
For this set of examples: A = 1 (low), B = 3000 (high)
Ratio of odd integers: 1/2
Ratio of even integers: 1/2
Ratio of integers divisible by 5 (N = 5): 1/5
Ratio of perfect squares: 9/500
Ratio of integers beginning with 2: 1111/3000
Ratio of triangular numbers: 19/750
Ratio of integers relatively prime to 250 (N = 250): 2/5
Source
Diaconis, Persi and Brian Skyrms Ten Great Ideas About Chance Princeton University Press: Princeton, NJ. 2018. ISBN 978-0-691-19639-8
Note: On June and July 2023, regular posts will be on Saturdays only. The Carnival of Math will be on June 3, 2023.
All original content copyright, © 2011-2023. 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.
