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