Lodumo transform

The Lodumo_m transform of a sequence s(n) (sometimes written lod_m s) is defined by a(n) = smallest nonnegative integer not yet in the list a such that a(n) ≡ s(n) (mod m). It is not a standard transformation from the mathematical literature, but is occasionally used in the OEIS. It was created by Philippe Deléham.

It creates a sequence without duplicates which is equivalent to the original sequence mod m. The resulting sequence is a permutation of the nonnegative integers if and only if all residue classes mod m appear infinitely often in the original sequence.

Maple

# @param L the input list of non-negative numbers 
# @param m  the remainder of the modulo operation. This is  only useful
#           if it's a number larger than 1.
# @return the sequence of nonnegative numbers a(n)
#
# Example: generate A160016 from A159833:
#     A159833 := [seq(n^2*(n^2+15)/4,n=0..100) ];
#     LODUMO(A159833,2) ;
#
# Example: generate A160081 from A000045:
#     A000045 := [seq(combinat[fibonacci](n),n=0..100) ];
#     LODUMO(A000045,5) ;
#
# Richard J. Mathar, 2009-04-30
LODUMO := proc(L,m)
    local a,n, an;
    if not type(L,'list') then
        ERROR("Expected list type argument instead of ",whattype(L) ) ;
    end if;
    a := [] ;
    for n from 1 to nops(L) do
        for an from op(n,L) mod m by m do
            if not an in a then
                a := [op(a),an] ;
                break;
            fi;
        od:
    end do;
    a ;
end proc:

Perl

sub lodumo { #  Georg Fischer, Feb 15 2019
    my ($m, @list) = @_;
    my @a = ();
    my $il = 0;
    while ($il < scalar(@list)) {
        my $busy = 1;
        my $an = $list[$il] % $m; 
        while ($busy == 1) {
            if (scalar(grep {$_ == $an} @a) == 0) {
                push(@a, $an);
                $busy = 0;
            }
            $an += $m;
        } # while $busy
        $il ++;
    } # while
    return @a;
} # sub lodumo

my @a = &lodumo($m, @list);
for (my $n = 0; $n < scalar(@list); $n ++) {
    print "$n $a[$n]\n";
} # for

Cite this as

Charles R Greathouse IV and Georg Fischer, Lodumo transform.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Lodumo_transform]