mirror of
https://github.com/Zeal-Operating-System/ZealOS.git
synced 2024-12-26 15:26:43 +00:00
375 lines
6.8 KiB
HolyC
Executable file
375 lines
6.8 KiB
HolyC
Executable file
/*The magic pairs problem:
|
|
|
|
Let SumFact(n) be the sum of factors
|
|
of n.
|
|
|
|
Find all n1,n2 in a range such that
|
|
|
|
SumFact(n1)-n1-1==n2 and
|
|
SumFact(n2)-n2-1==n1
|
|
|
|
-----------------------------------------------------
|
|
To find SumFact(k), start with prime factorization:
|
|
|
|
k=(p1^n1)(p2^n2) ... (pN^nN)
|
|
|
|
THEN,
|
|
|
|
SumFact(k)=(1+p1+p1^2...p1^n1)*(1+p2+p2^2...p2^n2)*
|
|
(1+pN+pN^2...pN^nN)
|
|
|
|
PROOF:
|
|
|
|
Do a couple examples -- it's obvious:
|
|
|
|
48=2^4*3
|
|
|
|
SumFact(48)=(1+2+4+8+16)*(1+3)=1+2+4+8+16+3+6+12+24+48
|
|
|
|
75=3*5^2
|
|
|
|
SumFact(75)=(1+3)*(1+5+25) =1+5+25+3+15+75
|
|
|
|
Corollary:
|
|
|
|
SumFact(k)=SumFact(p1^n1)*SumFact(p2^n2)*...*SumFact(pN^nN)
|
|
|
|
*/
|
|
|
|
//Primes are needed to sqrt(N). Therefore, we can use U32.
|
|
class PowPrime
|
|
{
|
|
I64 n;
|
|
I64 sumfact; //Sumfacts for powers of primes are needed beyond sqrt(N)
|
|
};
|
|
|
|
class Prime
|
|
{
|
|
U32 prime,pow_cnt;
|
|
PowPrime *pp;
|
|
};
|
|
|
|
I64 *PrimesNew(I64 N,I64 *_sqrt_primes,I64 *_cbrt_primes)
|
|
{
|
|
I64 i,j,sqrt=Ceil(Sqrt(N)),cbrt=Ceil(N`(1/3.0)),sqrt_sqrt=Ceil(Sqrt(sqrt)),
|
|
sqrt_primes=0,cbrt_primes=0;
|
|
U8 *s=CAlloc((sqrt+1+7)/8);
|
|
Prime *primes,*p;
|
|
|
|
for (i=2;i<=sqrt_sqrt;i++) {
|
|
if (!Bt(s,i)) {
|
|
j=i*2;
|
|
while (j<=sqrt) {
|
|
Bts(s,j);
|
|
j+=i;
|
|
}
|
|
}
|
|
}
|
|
for (i=2;i<=sqrt;i++)
|
|
if (!Bt(s,i)) {
|
|
sqrt_primes++; //Count primes
|
|
if (i<=cbrt)
|
|
cbrt_primes++;
|
|
}
|
|
|
|
p=primes=CAlloc(sqrt_primes*sizeof(Prime));
|
|
for (i=2;i<=sqrt;i++)
|
|
if (!Bt(s,i)) {
|
|
p->prime=i;
|
|
p++;
|
|
}
|
|
Free(s);
|
|
|
|
*_sqrt_primes=sqrt_primes;
|
|
*_cbrt_primes=cbrt_primes;
|
|
return primes;
|
|
}
|
|
|
|
PowPrime *PowPrimesNew(I64 N,I64 sqrt_primes,Prime *primes,I64 *_num_powprimes)
|
|
{
|
|
I64 i,j,k,sf,num_powprimes=0;
|
|
Prime *p;
|
|
PowPrime *powprimes,*pp;
|
|
|
|
p=primes;
|
|
for (i=0;i<sqrt_primes;i++) {
|
|
num_powprimes+=Floor(Ln(N)/Ln(p->prime));
|
|
p++;
|
|
}
|
|
|
|
p=primes;
|
|
pp=powprimes=MAlloc(num_powprimes*sizeof(PowPrime));
|
|
for (i=0;i<sqrt_primes;i++) {
|
|
p->pp=pp;
|
|
j=p->prime;
|
|
k=1;
|
|
sf=1;
|
|
while (j<N) {
|
|
sf+=j;
|
|
pp->n=j;
|
|
pp->sumfact=sf;
|
|
j*=p->prime;
|
|
pp++;
|
|
p->pow_cnt++;
|
|
}
|
|
p++;
|
|
}
|
|
*_num_powprimes=num_powprimes;
|
|
return powprimes;
|
|
}
|
|
|
|
I64 SumFact(I64 n,I64 sqrt_primes,Prime *p)
|
|
{
|
|
I64 i,k,sf=1;
|
|
PowPrime *pp;
|
|
if (n<2)
|
|
return 1;
|
|
for (i=0;i<sqrt_primes;i++) {
|
|
k=0;
|
|
while (!(n%p->prime)) {
|
|
n/=p->prime;
|
|
k++;
|
|
}
|
|
if (k) {
|
|
pp=p->pp+(k-1);
|
|
sf*=pp->sumfact;
|
|
if (n==1)
|
|
return sf;
|
|
}
|
|
p++;
|
|
}
|
|
return sf*(1+n); //Prime
|
|
}
|
|
|
|
Bool TestSumFact(I64 n,I64 target_sf,I64 sqrt_primes,I64 cbrt_primes,Prime *p)
|
|
{
|
|
I64 i=0,k,b,x1,x2;
|
|
PowPrime *pp;
|
|
F64 disc;
|
|
if (n<2)
|
|
return FALSE;
|
|
while (i++<cbrt_primes) {
|
|
k=0;
|
|
while (!(n%p->prime)) {
|
|
n/=p->prime;
|
|
k++;
|
|
}
|
|
if (k) {
|
|
pp=p->pp+(k-1);
|
|
if (ModU64(&target_sf,pp->sumfact))
|
|
return FALSE;
|
|
if (n==1) {
|
|
if (target_sf==1)
|
|
return TRUE;
|
|
else
|
|
return FALSE;
|
|
}
|
|
}
|
|
p++;
|
|
}
|
|
/* At this point we have three possible cases to test
|
|
1)n==p1 ->sf==(1+p1) ?
|
|
2)n==p1*p1 ->sf==(1+p1+p1^2) ?
|
|
3)n==p1*p2 ->sf==(p1+1)*(p2+1) ?
|
|
|
|
*/
|
|
if (1+n==target_sf) {
|
|
while (i++<sqrt_primes) {
|
|
k=0;
|
|
while (!(n%p->prime)) {
|
|
n/=p->prime;
|
|
k++;
|
|
}
|
|
if (k) {
|
|
pp=p->pp+(k-1);
|
|
if (ModU64(&target_sf,pp->sumfact))
|
|
return FALSE;
|
|
if (n==1) {
|
|
if (target_sf==1)
|
|
return TRUE;
|
|
else
|
|
return FALSE;
|
|
}
|
|
}
|
|
p++;
|
|
}
|
|
if (1+n==target_sf)
|
|
return TRUE;
|
|
else
|
|
return FALSE;
|
|
}
|
|
|
|
k=Sqrt(n);
|
|
if (k*k==n) {
|
|
if (1+k+n==target_sf)
|
|
return TRUE;
|
|
else
|
|
return FALSE;
|
|
} else {
|
|
// n==p1*p2 -> sf==(p1+1)*(p2+1) ? where p1!=1 && p2!=1
|
|
// if p1==1 || p2==1, it is FALSE because we checked a single prime above.
|
|
|
|
// sf==(p1+1)*(n/p1+1)
|
|
// sf==n+p1+n/p1+1
|
|
// sf*p1==n*p1+p1^2+n+p1
|
|
// p1^2+(n+1-sf)*p1+n=0
|
|
// x=(-b+/-sqrt(b^2-4ac))/2a
|
|
// a=1
|
|
// x=(-b+/-sqrt(b^2-4c))/2
|
|
// b=n+1-sf;c=n
|
|
b=n+1-target_sf;
|
|
// x=(-b+/-sqrt(b^2-4n))/2
|
|
disc=b*b-4*n;
|
|
if (disc<0)
|
|
return FALSE;
|
|
x1=(-b-Sqrt(disc))/2;
|
|
if (x1<=1)
|
|
return FALSE;
|
|
x2=n/x1;
|
|
if (x2>1 && x1*x2==n)
|
|
return TRUE;
|
|
else
|
|
return FALSE;
|
|
}
|
|
}
|
|
|
|
U0 PutFactors(I64 n) //For debugging
|
|
{
|
|
I64 i,k,sqrt=Ceil(Sqrt(n));
|
|
for (i=2;i<=sqrt;i++) {
|
|
k=0;
|
|
while (!(n%i)) {
|
|
k++;
|
|
n/=i;
|
|
}
|
|
if (k) {
|
|
"%d",i;
|
|
if (k>1)
|
|
"^%d",k;
|
|
'' CH_SPACE;
|
|
}
|
|
}
|
|
if (n!=1)
|
|
"%d ",n;
|
|
}
|
|
|
|
class RangeJob
|
|
{
|
|
CDoc *doc;
|
|
I64 num,lo,hi,N,sqrt_primes,cbrt_primes;
|
|
Prime *primes;
|
|
CJob *cmd;
|
|
} rj[mp_cnt];
|
|
|
|
I64 TestCoreSubRange(RangeJob *r)
|
|
{
|
|
I64 i,j,m,n,n2,sf,res=0,range=r->hi-r->lo,
|
|
*sumfacts=MAlloc(range*sizeof(I64)),
|
|
*residue =MAlloc(range*sizeof(I64));
|
|
U16 *pow_cnt =MAlloc(range*sizeof(U16));
|
|
Prime *p=r->primes;
|
|
PowPrime *pp;
|
|
MemSetI64(sumfacts,1,range);
|
|
for (n=r->lo;n<r->hi;n++)
|
|
residue[n-r->lo]=n;
|
|
for (j=0;j<r->sqrt_primes;j++) {
|
|
MemSet(pow_cnt,0,range*sizeof(U16));
|
|
m=1;
|
|
for (i=0;i<p->pow_cnt;i++) {
|
|
m*=p->prime;
|
|
n=m-r->lo%m;
|
|
while (n<range) {
|
|
pow_cnt[n]++;
|
|
n+=m;
|
|
}
|
|
}
|
|
for (n=0;n<range;n++)
|
|
if (i=pow_cnt[n]) {
|
|
pp=&p->pp[i-1];
|
|
sumfacts[n]*=pp->sumfact;
|
|
residue [n]/=pp->n;
|
|
}
|
|
p++;
|
|
}
|
|
|
|
for (n=0;n<range;n++)
|
|
if (residue[n]!=1)
|
|
sumfacts[n]*=1+residue[n];
|
|
|
|
for (n=r->lo;n<r->hi;n++) {
|
|
sf=sumfacts[n-r->lo];
|
|
n2=sf-n-1;
|
|
if (n<n2<r->N) {
|
|
if (r->lo<=n2<r->hi && sumfacts[n2-r->lo]-n2-1==n ||
|
|
TestSumFact(n2,sf,r->sqrt_primes,r->cbrt_primes,r->primes)) {
|
|
DocPrint(r->doc,"%u:%u\n",n,sf-n-1);
|
|
res++;
|
|
}
|
|
}
|
|
}
|
|
Free(pow_cnt);
|
|
Free(residue);
|
|
Free(sumfacts);
|
|
return res;
|
|
}
|
|
|
|
#define CORE_SUB_RANGE 0x1000
|
|
|
|
I64 TestCoreRange(RangeJob *r)
|
|
{
|
|
I64 i,n,res=0;
|
|
RangeJob rj;
|
|
MemCpy(&rj,r,sizeof(RangeJob));
|
|
for (i=r->lo;i<r->hi;i+=CORE_SUB_RANGE) {
|
|
rj.lo=i;
|
|
rj.hi=i+CORE_SUB_RANGE;
|
|
if (rj.hi>r->hi)
|
|
rj.hi=r->hi;
|
|
res+=TestCoreSubRange(&rj);
|
|
|
|
n=rj.hi-rj.lo;
|
|
lock {progress1+=n;}
|
|
|
|
Yield;
|
|
}
|
|
return res;
|
|
}
|
|
|
|
I64 MagicPairs(I64 N)
|
|
{
|
|
F64 t0=tS;
|
|
I64 res=0;
|
|
I64 sqrt_primes,cbrt_primes,num_powprimes,
|
|
i,k,n=(N-1)/mp_cnt+1;
|
|
Prime *primes=PrimesNew(N,&sqrt_primes,&cbrt_primes);
|
|
PowPrime *powprimes=PowPrimesNew(N,sqrt_primes,primes,&num_powprimes);
|
|
|
|
"N:%u SqrtPrimes:%u CbrtPrimes:%u PowersOfPrimes:%u\n",
|
|
N,sqrt_primes,cbrt_primes,num_powprimes;
|
|
progress1=0;
|
|
*progress1_desc=0;
|
|
progress1_max=N;
|
|
k=2;
|
|
for (i=0;i<mp_cnt;i++) {
|
|
rj[i].doc=DocPut;
|
|
rj[i].num=i;
|
|
rj[i].lo=k;
|
|
k+=n;
|
|
if (k>N) k=N;
|
|
rj[i].hi=k;
|
|
rj[i].N=N;
|
|
rj[i].sqrt_primes=sqrt_primes;
|
|
rj[i].cbrt_primes=cbrt_primes;
|
|
rj[i].primes=primes;
|
|
rj[i].cmd=JobQue(&TestCoreRange,&rj[i],mp_cnt-1-i,0);
|
|
}
|
|
for (i=0;i<mp_cnt;i++)
|
|
res+=JobResGet(rj[i].cmd);
|
|
Free(powprimes);
|
|
Free(primes);
|
|
"Found:%u Time:%9.4f\n",res,tS-t0;
|
|
progress1=progress1_max=0;
|
|
return res;
|
|
}
|
|
|
|
MagicPairs(1000000);
|