Integer Ratios and Integer Density
03-22-2023, 03:14 AM
Post: #1
 Eddie W. Shore Senior Member Posts: 1,584 Joined: Dec 2013
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:

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
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