How to resolve the algorithm Numbers which are the cube roots of the product of their proper divisors step by step in the PL/M programming language
Published on 12 May 2024 09:40 PM
How to resolve the algorithm Numbers which are the cube roots of the product of their proper divisors step by step in the PL/M programming language
Table of Contents
Problem Statement
Consider the number 24. Its proper divisors are: 1, 2, 3, 4, 6, 8 and 12. Their product is 13,824 and the cube root of this is 24. So 24 satisfies the definition in the task title. Compute and show here the first 50 positive integers which are the cube roots of the product of their proper divisors. Also show the 500th and 5,000th such numbers. Compute and show the 50,000th such number. OEIS considers 1 to be the first number in this sequence even though, strictly speaking, it has no proper divisors. Please therefore do likewise.
Let's start with the solution:
Step by Step solution about How to resolve the algorithm Numbers which are the cube roots of the product of their proper divisors step by step in the PL/M programming language
Source code in the pl/m programming language
100H: /* FIND NUMBERS THAT ARE THE CUBE ROOT OF THEIR PROPER DIVISORS */
DECLARE FALSE LITERALLY '0', TRUE LITERALLY '0FFH';
/* CP/M SYSTEM CALL AND I/O ROUTINES */
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
PR$CHAR: PROCEDURE( C ); DECLARE C BYTE; CALL BDOS( 2, C ); END;
PR$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S ); END;
PR$NL: PROCEDURE; CALL PR$CHAR( 0DH ); CALL PR$CHAR( 0AH ); END;
PR$NUMBER: PROCEDURE( N ); /* PRINTS A NUMBER IN THE MINIMUN FIELD WIDTH */
DECLARE N ADDRESS;
DECLARE V ADDRESS, N$STR ( 6 )BYTE, W BYTE;
V = N;
W = LAST( N$STR );
N$STR( W ) = '$';
N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL PR$STRING( .N$STR( W ) );
END PR$NUMBER;
/* END SYSTEM CALL AND I/O ROUTINES */
DECLARE PDC ( 5000 )ADDRESS;
DECLARE ( I, I2, J, COUNT ) ADDRESS;
DO I = 1 TO LAST( PDC ); PDC( I ) = 1; END;
DO I = 2 TO LAST( PDC );
I2 = I + I;
DO J = I2 TO LAST( PDC ) BY I;
PDC( J ) = PDC( J ) + 1;
END;
END;
PDC( 1 ) = 7;
COUNT, I = 0;
DO WHILE COUNT < 500 AND I < LAST( PDC );
I = I + 1;
IF PDC( I ) = 7 THEN DO;
IF ( COUNT := COUNT + 1 ) < 51 THEN DO;
CALL PR$CHAR( ' ' );
IF I < 10 THEN CALL PR$CHAR( ' ' );
IF I < 100 THEN CALL PR$CHAR( ' ' );
IF I < 1000 THEN CALL PR$CHAR( ' ' );
CALL PR$NUMBER( I );
IF COUNT MOD 10 = 0 THEN CALL PR$NL;
END;
ELSE IF COUNT = 500 THEN DO;
CALL PR$NUMBER( COUNT );
CALL PR$STRING( .'TH: $' );
CALL PR$NUMBER( I );
CALL PR$NL;
END;
END;
END;
EOF
100H: /* FIND NUMBERS THAT ARE THE CUBE ROOT OF THEIR PROPER DIVISORS */
DECLARE FALSE LITERALLY '0', TRUE LITERALLY '0FFH';
/* CP/M SYSTEM CALL AND I/O ROUTINES */
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
PR$CHAR: PROCEDURE( C ); DECLARE C BYTE; CALL BDOS( 2, C ); END;
PR$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S ); END;
PR$NL: PROCEDURE; CALL PR$CHAR( 0DH ); CALL PR$CHAR( 0AH ); END;
PR$NUMBER: PROCEDURE( N ); /* PRINTS A NUMBER IN THE MINIMUN FIELD WIDTH */
DECLARE N ADDRESS;
DECLARE V ADDRESS, N$STR ( 6 )BYTE, W BYTE;
V = N;
W = LAST( N$STR );
N$STR( W ) = '$';
N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL PR$STRING( .N$STR( W ) );
END PR$NUMBER;
/* END SYSTEM CALL AND I/O ROUTINES */
DECLARE MAX$PF LITERALLY '200';
/* SETS PF$A AND PFC$A TO THE PRIME FACTORS AND COUNTS OF F, THE NUMBER */
/* NUMBER OF FACTORS IS RETURNED IN PF$POS$PTR */
/* PF$POS$PTR MUST BE INITIALISED BEFORE THE CALL */
FACTORISE: PROCEDURE( F, PF$POS$PTR, PF$A, PFC$A );
DECLARE ( F, PF$POS$PTR, PF$A, PFC$A ) ADDRESS;
DECLARE PF$POS BASED PF$POS$PTR ADDRESS;
DECLARE PF BASED PF$A ( 0 )ADDRESS;
DECLARE PFC BASED PFC$A ( 0 )ADDRESS;
DECLARE ( FF, V, POWER ) ADDRESS;
/* START WITH 2 */
V = F;
FF = 2;
DO WHILE V > 1;
IF V MOD FF = 0 THEN DO;
/* FF IS A PRIME FACTOR OF V */
DECLARE P ADDRESS;
POWER = 0;
DO WHILE V MOD FF = 0;
POWER = POWER + 1;
V = V / FF;
END;
P = 0;
DO WHILE P < PF$POS AND PF( P ) <> FF;
P = P + 1;
END;
IF P >= PF$POS THEN DO;
/* FIRST TIME FF HAS APPEARED AS A PRIME FACTOR */
P = PF$POS;
PFC( P ) = 0;
PF$POS = PF$POS + 1;
END;
PF( P ) = FF;
PFC( P ) = PFC( P ) + POWER;
END;
IF FF = 2 THEN FF = 3; ELSE FF = FF + 2;
END;
END FACTORISE;
/* RETURNS TRUE THE PRODUCT OF THE PROPER DIVISORS OF N IS THE CUBE OF N */
/* FALSE OTHERWISE */
PD$PRODUCT$IS$CUBE: PROCEDURE( N )ADDRESS;
DECLARE N ADDRESS;
DECLARE IS$CUBE BYTE;
IF N < 2
THEN IS$CUBE = TRUE;
ELSE DO;
DECLARE ( I, PF$POS, NF$POS ) ADDRESS;
DECLARE ( PF, PFC, NF, NFC ) ( MAX$PF ) ADDRESS;
PFC( 0 ), PF( 0 ), PF$POS, NFC( 0 ), NF( 0 ), NF$POS = 0;
/* FACTORISE N */
CALL FACTORISE( N, .NF$POS, .NF, .NFC );
/* COPY FACTORS BUT ZERO THE COUNTS SO WE CAN EASILY CHECK THE */
/* FACTORS OF N ARE THE SAME AS THOSE OF THE PROPER DIVISOR PRODUCT */
DO I = 0 TO NF$POS - 1;
PF( I ) = NF( I );
PFC( I ) = 0;
END;
/* FIND THE PROPER DIVISORS AND FACTORISE THEM, ACCUMULATING THE */
/* PRIME FACTOR COUNTS */
I = 2;
DO WHILE I * I <= N;
IF N MOD I = 0 THEN DO;
/* I IS A DIVISOR OF N */
DECLARE ( F, G ) ADDRESS;
F = I; /* FIRST FACTOR */
G = N / F; /* SECOND FACTOR */
/* FACTORISE F, COUNTING THE PRIME FACTORS */
CALL FACTORISE( F, .PF$POS, .PF, .PFC );
/* FACTORISE G, IF IT IS NOT THE SAME AS F */
IF F <> G THEN CALL FACTORISE( G, .PF$POS, .PF, .PFC );
END;
I = I + 1;
END;
IS$CUBE = PF$POS = NF$POS;
IF IS$CUBE THEN DO;
/* N AND ITS PROPER DIVISOR PRODUCT HAVE THE SAME PRIME FACTOR */
/* COUNT - CHECK THE PRIME FACTLORS ARE THE SAME AND THAT THE */
/* PRODUCTS FACTORS APPEAR 3 TIMEs THOSE OF N */
I = 0;
DO WHILE I < PF$POS AND IS$CUBE;
IS$CUBE = ( PF( I ) = NF( I ) )
AND ( PFC( I ) = NFC( I ) * 3 );
I = I + 1;
END;
END;
END;
RETURN IS$CUBE;
END;
/* RETURNS THE PROPER DIVISOR PRODUCT OF N, MOD 65536 */
PDP: PROCEDURE( N )ADDRESS;
DECLARE N ADDRESS;
DECLARE ( I, I2, PRODUCT ) ADDRESS;
PRODUCT = 1;
I = 2;
DO WHILE ( I2 := I * I ) <= N;
IF N MOD I = 0 THEN DO;
PRODUCT = PRODUCT * I;
IF I2 <> N THEN DO;
PRODUCT = PRODUCT * ( N / I );
END;
END;
I = I + 1;
END;
RETURN PRODUCT;
END PDP;
DECLARE ( I, I3, J, COUNT ) ADDRESS;
COUNT, I = 0;
DO WHILE COUNT < 5$000;
I = I + 1;
I3 = I * I * I;
IF PDP( I ) = I3 THEN DO;
/* THE PROPER DIVISOR PRODUCT MOD 65536 IS THE SAME AS N CUBED ALSO */
/* MOD 65536, IF THE CUBE IS 0 MOD 65536, WE NEED TO CHECK THE */
/* PRIME FACTORS */
DECLARE IS$NUMBER BYTE;
IF I3 <> 0 THEN IS$NUMBER = TRUE;
ELSE IS$NUMBER = PD$PRODUCT$IS$CUBE( I );
IF IS$NUMBER THEN DO;
IF ( COUNT := COUNT + 1 ) < 51 THEN DO;
CALL PR$CHAR( ' ' );
IF I < 10 THEN CALL PR$CHAR( ' ' );
IF I < 100 THEN CALL PR$CHAR( ' ' );
IF I < 1000 THEN CALL PR$CHAR( ' ' );
CALL PR$NUMBER( I );
IF COUNT MOD 10 = 0 THEN CALL PR$NL;
END;
ELSE IF COUNT = 500 OR COUNT = 5000 THEN DO;
IF COUNT < 1000 THEN CALL PR$CHAR( ' ' );
CALL PR$STRING( .' $' );
CALL PR$NUMBER( COUNT );
CALL PR$STRING( .'TH: $' );
CALL PR$NUMBER( I );
CALL PR$NL;
END;
END;
END;
END;
EOF
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