How to resolve the algorithm Duffinian numbers step by step in the MAD programming language

Published on 12 May 2024 09:40 PM
#Mad

How to resolve the algorithm Duffinian numbers step by step in the MAD programming language

Table of Contents

Problem Statement

A Duffinian number is a composite number k that is relatively prime to its sigma sum σ. The sigma sum of k is the sum of the divisors of k.

161 is a Duffinian number.

Duffinian numbers are very common. It is not uncommon for two consecutive integers to be Duffinian (a Duffinian twin) (8, 9), (35, 36), (49, 50), etc. Less common are Duffinian triplets; three consecutive Duffinian numbers. (63, 64, 65), (323, 324, 325), etc. Much, much less common are Duffinian quadruplets and quintuplets. The first Duffinian quintuplet is (202605639573839041, 202605639573839042, 202605639573839043, 202605639573839044, 202605639573839045). It is not possible to have six consecutive Duffinian numbers

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Duffinian numbers step by step in the MAD programming language

Source code in the mad programming language

            NORMAL MODE IS INTEGER
            DIMENSION SIGMA(10000),OUTROW(10)

            INTERNAL FUNCTION(AA,BB)
            ENTRY TO GCD.
            A = AA
            B = BB
STEP        WHENEVER A.E.B, FUNCTION RETURN A
            WHENEVER A.G.B, A = A-B
            WHENEVER A.L.B, B = B-A
            TRANSFER TO STEP
            END OF FUNCTION

            INTERNAL FUNCTION(N)
            ENTRY TO DUFF.
            SIG = SIGMA(N)
            FUNCTION RETURN SIG.G.N+1 .AND. GCD.(N,SIG).E.1
            END OF FUNCTION

            INTERNAL FUNCTION(N)
            ENTRY TO TRIP.
            FUNCTION RETURN DUFF.(N) .AND.
          0      DUFF.(N+1) .AND. DUFF.(N+2)
            END OF FUNCTION

            THROUGH SZERO, FOR I=1, 1, I.G.10000
SZERO       SIGMA(I) = 0
            THROUGH SCALC, FOR I=1, 1, I.G.10000
            THROUGH SCALC, FOR J=I, I, J.G.10000
SCALC       SIGMA(J) = SIGMA(J) + I

            PRINT COMMENT $ FIRST 50 DUFFINIAN NUMBERS$
            CAND = 0
            THROUGH DUFROW, FOR R=0, 1, R.GE.5
            THROUGH DUFCOL, FOR C=0, 1, C.GE.10
SCHDUF      THROUGH SCHDUF, FOR CAND=CAND+1, 1, DUFF.(CAND)
DUFCOL      OUTROW(C) = CAND
DUFROW      PRINT FORMAT ROWFMT,OUTROW(0),OUTROW(1),OUTROW(2),
          0           OUTROW(3),OUTROW(4),OUTROW(5),OUTROW(6),
          1           OUTROW(7),OUTROW(8),OUTROW(9)

            PRINT COMMENT $ $
            PRINT COMMENT $ FIRST 15 DUFFINIAN TRIPLETS$
            CAND = 0
            THROUGH DUFTRI, FOR S=0, 1, S.GE.15
SCHTRP      THROUGH SCHTRP, FOR CAND=CAND+1, 1, TRIP.(CAND)
DUFTRI      PRINT FORMAT TRIFMT,CAND,CAND+1,CAND+2

            VECTOR VALUES ROWFMT = $10(I5)*$
            VECTOR VALUES TRIFMT = $3(I7)*$
            END OF PROGRAM

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