**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.