<Section Label="sec-integers">
<Heading>Functions for integers</Heading>
<ManSection>
<Func Name="AllSmoothIntegers"
Arg="maxp maxn" Label="for two integers" />
<Func Name="AllSmoothIntegers"
Arg="L maxp" Label="for a list and an integer" />
<Description>
This function has been transferred from package &RCWA;.
<P/>
<Index>smooth integer</Index>
The function <C>AllSmoothIntegers(<A>maxp</A>,<A>maxn</A>)</C>
returns the list of all positive integers less than or equal to <A>maxn</A>
whose prime factors are all in the list
<M>L = \{p ~|~ p \leqslant maxp, p~\mbox{prime} \}</M>.
<P/>
In the alternative form, when <M>L</M> is a list of primes,
the function returns the list of all positive integers
whose prime factors lie in <M>L</M>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> AllSmoothIntegers( 3, 1000 );
[ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96,
108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576,
648, 729, 768, 864, 972 ]
gap> AllSmoothIntegers( [5,11,17], 1000 );
[ 1, 5, 11, 17, 25, 55, 85, 121, 125, 187, 275, 289, 425, 605, 625, 935 ]
gap> Length( last );
16
gap> List( [3..20], n -> Length( AllSmoothIntegers( [5,11,17], 10^n ) ) );
[ 16, 29, 50, 78, 114, 155, 212, 282, 359, 452, 565, 691, 831, 992, 1173,
1374, 1595, 1843 ]
]]>
</Example>
<ManSection>
<Func Name="AllProducts"
Arg="L, k" />
<Description>
This function has been transferred from package &RCWA;.
<P/>
The command <C>AllProducts(<A>L</A>,<A>k</A>)</C> returns the list of
all products of <A>k</A> entries of the list <A>L</A>.
Note that every ordering of the entries is used so that, in the commuting case,
there are bound to be repetitions.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> AllProducts([1..4],3);
[ 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12,
16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27,
36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32,
48, 64 ]
gap> Set(last);
[ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 64 ]
gap> AllProducts( [(1,2,3),(2,3,4)], 2 );
[ (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,3,2) ]
]]>
</Example>
<ManSection>
<Func Name="RestrictedPartitionsWithoutRepetitions"
Arg="n,S" />
<Description>
This function has been transferred from package &RCWA;.
<P/>
For a positive integer <A>n</A> and a set of positive integers <A>S</A>,
this function returns the list of partitions of <A>n</A>
into distinct elements of <A>S</A>.
Unlike <C>RestrictedPartitions</C>, no repetitions are allowed.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> RestrictedPartitions( 20, [4..10] );
[ [ 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5 ], [ 6, 5, 5, 4 ], [ 6, 6, 4, 4 ],
[ 7, 5, 4, 4 ], [ 7, 7, 6 ], [ 8, 4, 4, 4 ], [ 8, 6, 6 ], [ 8, 7, 5 ],
[ 8, 8, 4 ], [ 9, 6, 5 ], [ 9, 7, 4 ], [ 10, 5, 5 ], [ 10, 6, 4 ],
[ 10, 10 ] ]
gap> RestrictedPartitionsWithoutRepetitions( 20, [4..10] );
[ [ 10, 6, 4 ], [ 9, 7, 4 ], [ 9, 6, 5 ], [ 8, 7, 5 ] ]
gap> RestrictedPartitionsWithoutRepetitions( 10^2, List([1..10], n->n^2 ) );
[ [ 100 ], [ 64, 36 ], [ 49, 25, 16, 9, 1 ] ]
]]>
</Example>
<ManSection>
<Func Name="NextProbablyPrimeInt"
Arg="n" />
<Description>
This function has been transferred from package &RCWA;.
<P/>
The function <C>NextProbablyPrimeInt(<A>n</A>)</C>
does the same as <C>NextPrimeInt(<A>n</A>)</C>
except that for reasons of performance it tests numbers only for
<C>IsProbablyPrimeInt(<A>n</A>)</C> instead of <C>IsPrimeInt(<A>n</A>)</C>.
For large <A>n</A>, this function is much faster than
<C>NextPrimeInt(<A>n</A>)</C>
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> n := 2^251;
3618502788666131106986593281521497120414687020801267626233049500247285301248
gap> NextProbablyPrimeInt( n );
3618502788666131106986593281521497120414687020801267626233049500247285301313
gap> time;
1
gap> NextPrimeInt( n );
3618502788666131106986593281521497120414687020801267626233049500247285301313
gap> time;
213
]]>
</Example>
<ManSection>
<Func Name="PrimeNumbersIterator"
Arg="[chunksize]" />
<Description>
This function has been transferred from package &RCWA;.
<P/>
This function returns an iterator which runs over the prime numbers
n ascending order; it takes an optional argument <C>chunksize</C>
which specifies the length of the interval which is sieved in one go
(the default is <M>10^7</M>),
and which can be used to balance runtime vs. memory consumption.
It is assumed that <C>chunksize</C> is larger than any gap between two
consecutive primes within the range one intends to run the iterator over.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> iter := PrimeNumbersIterator();;
gap> for i in [1..100] do p := NextIterator(iter); od;
gap> p;
541
gap> sum := 0;;
gap> ## "prime number race" 1 vs. 3 mod 4
gap> for p in PrimeNumbersIterator() do
> if p <> 2 then sum := sum + E(4)^(p-1); fi;
> if sum > 0 then break; fi;
> od;
gap> p;
26861
]]>
</Example>
</Section>
</Chapter>
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