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# generate all perfect numbers until 100. ...

## generate all perfect numbers until 100. ...

(OP)
Someone asked how to calculate the perfect numbers until 100 .And know I don't find that person. But here it is

%generate all perfect number until 100...
%Create perfect numbers

%divisible(10,2).
divisible(X,Y):- N is Y*Y,N =< X,X mod Y =:= 0.
divisible(X,Y):- Y < X, Y1 is Y + 1, divisible(X,Y1).

%isprime([3],Z).
isprime([X|_],X):-Y is 2, X >1, \+divisible(X,Y).
isprime([_|T],Z):-isprime(T,Z).

%Calculate the power of one number
%ie power(2,10,R)
power(_,0,1):-!.
power(N,K,R):-K1 is K-1,
power(N,K1,R1),
R is R1*N.

%formula of perfect numbers 2^(p-1)*(2^p-1)
%ie calc(2,10,R)
calc(2,K,R):-power(2,K,X),R1 is X-1,
power(2,K-1,R2),
R is R1 * R2.

%using lists
%ie calc([2,3,4],R).
listperf([K|_],R):-calc(2,K,R).
listperf([_|T],Z):-listperf(T,Z).

%generate one list of N numbers.
%genList(10,L).
generateList(0,[]).
generateList(N,[X|Xs]):- N > 0,
X is N+1,
N1 is N-1,generateList(N1,Xs).

%list of N perfect numbers
%perfect(100,C)
perfect(N,C):-generateList(N,R),findall(L,isprime(R,L),P),listperf(P,C).

### RE: generate all perfect numbers until 100. ...

#### Quote:

C = 35184367894528 ;
C = 137438691328 ;
C = 8589869056 ;
C = 33550336 ;
C = 2096128 ;
C = 8128 ;
C = 496 ;
C = 28 ;
C = 6 ;
see here http://en.wikipedia.org/wiki/Perfect_number.
I think that the person who wanted to calculate perfect numbers until 100 was waiting for something like that :

#### CODE

perfect(N, L) :-
numlist(2, N, L1),
include(perfect, L1, L).

perfect(N) :-
divisors(N, 2, [1], L),
sumlist(L, N).

divisors(N, D, LC, LF) :-
0 is N mod D,
!,
D1 is D + 1,
X is N / D,
(   X >= D -> divisors(N, D1, [X, D |LC], LF); LC = LF).

divisors(N, D, LC, LF) :-
X is N / D,
(   X > D -> D1 is D + 1, divisors(N, D1, LC, LF); LF = LC).
This code use the definition of perfect numbers, it works for small numbers, for big numbers, I don't know if there is better algorithm.

### RE: generate all perfect numbers until 100. ...

(OP)
thanks

yes
I have one error on the first part of this formula

I need to check, not only that p is prime,but that 2^p-1 is also prime(first part of the formula).

### RE: generate all perfect numbers until 100. ...

(OP)
correction of last program.
This program need to improve performance with a prime numbers test faster.

%----------------
%Find perfect numbers

hasFactor(_,0):-!,fail.
hasFactor(N,L):-L<N,N mod L=:=0.
hasFactor(N,L):-L<N,L2 is L+1,hasFactor(N,L2).

%test of prime number
%isprime([3],Z).
isprime([X|_],X):-Y is 2, X >1, \+hasFactor(X,Y).
isprime([_|T],Z):-isprime(T,Z).

%multiply with sums
%product of natural numbers with successives sums:

%            |0              se y=0
%    mul(x,y)|
%            |x+mul(x,y-1)   se y>0

mul(_,0,0):-!.
mul(X,Y,Z):-X>0,Y>0,
Y1 is Y-1,
mul(X,Y1,Z1),Z is X+Z1.

%Calculate the power of one number
%ie power(2,10,R)
power(_,0,1):-!.
power(N,K,R):-K1 is K-1,
power(N,K1,R1),
mul(R1,N,R).

%generate one list of N numbers.
%genList(10,L).
generateList(0,[]).
generateList(N,[X|Xs]):- N > 0,
X is N+1,
N1 is N-1,generateList(N1,Xs).

%formula of perfect numbers (2^p-1)*2^(p-1)
%ie calc(2,10,R)
calc(2,K,R):-power(2,K,X),R1 is X-1,primo(R1),
power(2,K-1,R2),
mul(R1,R2,R).

primo(X):-Y is 2, X >1, \+hasFactor(X,Y).

%output the list
noLst([X|_],X).
noLst([_|T],Z):-noLst(T,Z).

%calculate perfect numbers
%ie perfect(12,N).
perfect(N,R):-generateList(N,Xs),noLst(Xs,K),calc(2,K,R).

### RE: generate all perfect numbers until 100. ...

As you have already been told in another forum, if you have predicate sqrt, hasFactor can be written

#### CODE

hasFactor(N,L):-
N mod L=:=0.

hasFactor(N,L):-
L<sqrt(N),
L2 is L+1,
hasFactor(N,L2).

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