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powerful.c
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#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "ptypes.h"
#define FUNC_isqrt 1
#define FUNC_ctz 1
#define FUNC_gcd_ui 1
#define FUNC_ipow 1
#include "util.h"
#include "sieve.h"
#include "cache.h"
#include "constants.h"
#include "inverse_interpolate.h"
int is_powerful(UV n, UV k) {
UV pk;
int res;
if (n <= 1) return (n==1);
if (k <= 1) return 1;
if (!(n&1)) { /* Check and remove all multiples of 2 */
if (n & ((UVCONST(1) << k)-1)) return 0;
n >>= ctz(n);
if (n == 1) return 1;
}
if (k > MPU_MAX_POW3) return 0;
/* if (k > logint(n,3)) return 0; */
/* Quick checks */
if (k == 2) {
if ( (!(n % 3) && (n % 9))
|| (!(n % 5) && (n % 25))
|| (!(n % 7) && (n % 49))
|| (!(n % 11) && (n % 121))
|| (!(n % 13) && (n % 169)) ) return 0;
} else if (k == 3) {
if ( (!(n % 3) && (n % 27))
|| (!(n % 5) && (n % 125))
|| (!(n % 7) && (n % 343))
|| (!(n % 11) && (n % 1331)) ) return 0;
} else {
if ( (!(n % 3) && (n % 81))
|| (!(n % 5) && (n % 625))
|| (!(n % 7) && (n % 2401))
|| (!(n % 11) && (n % 14641)) ) return 0;
}
#if 0 /* Full factoring. Very simple and reasonably efficient. */
{
UV fac[MPU_MAX_FACTORS+1];
UV exp[MPU_MAX_FACTORS+1];
int i, nfactors;
nfactors = factor_exp(n, fac, exp);
for (i = 0; i < nfactors; i++)
if (exp[i] < k)
return 0;
return 1;
}
#endif
/* Rather than full factoring, we'll use trial division. For k=2, we
* only need to check up to the fourth root of n, and k=3 to the sixth.
* Even for k=2 this is faster than full factoring on average. */
/* At every checkpoint (new prime p) for k=2 either:
* 1) N < p^4 and N=1, p^2, q^2, p^3, or q^3. (q>p). Return 1.
* 2) N < p^4 otherwise, N cannot be powerful. Return 0;
* 3) N = p^4 is_square caught this and returned 1.
* So the next possibility is p^2 * q^2 where q = next_prime(p).
* Check n < p^4 before each new prime, and condition 1 after modifying n.
*/
if (n == 1 || powerof(n) >= k) return 1;
#define LCHECK_POWERFUL(n, p, k) \
pk = ipow(p,k); \
if (n < pk*pk) { res = 0; break; } \
if (!(n%p)) { \
if (n%pk) { res = 0; break; } \
for (n /= pk; (n%p) == 0; n /= p) ; \
if (n == 1 || powerof(n) >= k) { res = 1; break; } \
}
#define CHECK_POWERFUL(n, p, k) \
do { LCHECK_POWERFUL(n, p, k); } while (0); \
if (res != -1) return res;
res = -1;
CHECK_POWERFUL(n, 3, k); if (k >= 14 || n < ipow( 5,2*k)) return 0;
CHECK_POWERFUL(n, 5, k); if (k >= 12 || n < ipow( 7,2*k)) return 0;
CHECK_POWERFUL(n, 7, k); if (k >= 10 || n < ipow(11,2*k)) return 0;
START_DO_FOR_EACH_PRIME(11, rootint(n,2*k)) {
LCHECK_POWERFUL(n, p, k);
} END_DO_FOR_EACH_PRIME
return (res == 1);
}
static UV _pcr(UV n, UV k, unsigned char* isf, UV m, UV r) {
UV i, sum = 0, lim = rootint(n/m, r);
if (r <= k) return lim;
if (r-1 == k) {
for (i = 1; i <= lim; i++)
if (isf[i] && gcd_ui(m,i) == 1)
sum += rootint(n/(m*ipow(i,r)),k);
} else {
for (i = 1; i <= lim; i++)
if (isf[i] && gcd_ui(m,i) == 1)
sum += _pcr(n, k, isf, m * ipow(i,r), r-1);
}
return sum;
}
UV powerful_count(UV n, UV k) {
UV i, r, lim, sum = 0;
unsigned char *isf;
if (k <= 1 || n <= 1) return n;
if (k >= BITS_PER_WORD) return 1;
lim = rootint(n, k+1);
isf = range_issquarefree(0, lim); /* in util.c */
if (k == 2) {
for (i = 1; i <= lim; i++)
if (isf[i])
sum += isqrt(n/(i*i*i));
} else {
/* sum = _pcr(n, k, isf, 1, 2*k-1); */
r = 2*k-1;
lim = rootint(n, r);
for (i = 1; i <= lim; i++)
if (isf[i])
sum += _pcr(n, k, isf, ipow(i,r), r-1);
}
Safefree(isf);
return sum;
}
/* We want:
* k=0 turned into k=2 in XS (0 here ok)
* n=0 undef in XS (0 here ok)
* k=1 => n
* n=1 => 1
* n=2 => 1<<k
* overflow here should return 0
*/
UV nth_powerful(UV n, UV k) {
static unsigned char const mink[20+1] = {0,0,1,2,4,6,7,9,11,12,14,16,18,19,21,23,24,26,28,30,31};
#if BITS_PER_WORD == 64
static UV const maxpow[11] = {0,UV_MAX,9330124695,11938035,526402,85014,25017,10251,5137,2903,1796};
#else
static UV const maxpow[11] = {0,UV_MAX,140008,6215,1373,536,281,172,115,79,57};
#endif
UV lo, hi, max;
double nc, npow, nest, dlo, dhi;
if (k == 0 || k >= BITS_PER_WORD) return 0;
if (k == 1 || n <= 1) return n;
max = (k <= 10) ? maxpow[k] : powerful_count(UV_MAX,k);
if (n > max) return 0;
if (n <= 20 && k >= mink[n]) return UVCONST(1) << (k+(n-2));
/* Now k >= 2, n >= 4 */
nc = pow(n, 2) / pow(2.1732543125195541, 2);
if (k == 2) { /* From Mincu and Panaitopol 2009 */
npow = pow(n, 5.0/3.0);
dlo = nc + 0.3 * npow;
dhi = nc + 0.5 * npow;
lo = (UV) dlo;
hi = (n < 170) ? 8575 : (dhi >= UV_MAX) ? UV_MAX : 1 + (UV) dhi;
} else if (k == 3) {
/* Splitting the range is hacky but overall this isn't bad */
if (n < 84000) {
nest = .06003 * pow(n, 2.865);
dlo = 0.96 * (nc + nest);
dhi = 1.08 * (nc + nest);
} else {
nest = .02209 * pow(n, 2.955);
dlo = 0.987 * (nc + nest);
dhi = 1.020 * (nc + nest);
}
lo = (UV) dlo;
if (n < 900) dhi *= 1.3;
if (n < 160) dhi = 1.3 * dhi + 600;
hi = (dhi >= UV_MAX) ? UV_MAX : 1 + (UV) dhi;
} else if (k <= 10) {
/* Slopppy but better than linear. 4 <= k <= 10. */
if (n < 200) {
npow = pow(n, 3.031 + 0.460*(k-4));
nest = (.5462 / pow(1.15, k-4)) * npow;
dlo = 0.51 * (nc + nest);
dhi = 1.86 * (nc + nest);
} else {
npow = pow(n, 3.690 + 0.665*(k-4));
nest = (.01275 / pow(4.11, k-4)) * npow;
dlo = 0.70 * (nc + nest);
dhi = 4.3 * (nc + nest);
}
lo = (UV) dlo;
hi = (dhi >= UV_MAX) ? UV_MAX : 1 + (UV) dhi;
} else {
lo = (UVCONST(1) << (k+1))+1;
hi = UV_MAX;
/* Linear from min to max rather than a nice power fit as above */
if (n < max) hi = lo + (((double)n / (double)max) * (UV_MAX-lo) + 1);
/* Alternately hi=0 which makes interpolation routine handle it. */
}
return inverse_interpolate_k(lo, hi, n, k, &powerful_count, 0);
}
static UV _sumpowerful(UV m, UV r, UV n, UV k, unsigned char* isf) {
UV i, rootdiv, sum;
if (r < k) return m;
rootdiv = rootint(n/m, r);
if (r == k) return m * powersum(rootdiv, r);
for (sum = 0, i = 1; i <= rootdiv; i++)
if (isf[i] && gcd_ui(i,m) == 1)
sum += _sumpowerful(m * ipow(i,r), r-1, n, k, isf);
return sum;
}
UV sumpowerful(UV n, UV k)
{
UV lim, sum;
unsigned char *isf;
#if BITS_PER_WORD == 64
static UV const maxpow[41] = {0,6074000999,8676161894447,263030040471727,
1856371767674975,6768543273131775,17199267839999999,35098120384607510,
62985599999999999,104857599999999999,157641800537109374,246512345193381887,
312499999999999999,406381963906121727,499999999999999999,592297667290202111,
701982420492091391,935976560656121855,1184595334580404223,1350851717672992088,
1579460446107205631,2105947261476274175,2369190669160808447,
4052555153018976266,4738381338321616895,7450580596923828124,
7450580596923828124,7450580596923828124,
UVCONST(9223372036854775807),UVCONST(9223372036854775807),
UVCONST(9223372036854775807),UVCONST(9223372036854775807),
UVCONST(9223372036854775807),UVCONST(9223372036854775807),
UVCONST(9223372036854775807),UVCONST(9223372036854775807),
UVCONST(9223372036854775807),UVCONST(9223372036854775807),
UVCONST(9223372036854775807),UVCONST(9223372036854775807),
UVCONST(12157665459056928800)};
if (k == 0 || n == 0) return 0;
if (k >= 64) return 1;
if (k < 41 && n > maxpow[k]) return 0;
#else
static UV const maxpow[21] = {0,92681,3367224,18224999,48599999,102036671,161243135,244140624,362797055,536870911,725594111,1088391167,1220703124,1220703124,2147483647,2147483647,2147483647,2147483647,2147483647,2147483647,3486784400};
if (k == 0 || n == 0) return 0;
if (k >= 32) return 1;
if (k < 21 && n > maxpow[k]) return 0;
#endif
if (k == 1) return (n & 1) ? n*((n+1)>>1) : (n>>1)*(n+1);
lim = rootint(n, k+1);
isf = range_issquarefree(0, lim);
sum = _sumpowerful(1, 2*k-1, n, k, isf);
Safefree(isf);
return sum;
}
static void _pcg(UV lo, UV hi, UV k, UV m, UV r, UV *pn, UV *npn)
{
UV v, *pnptr = pn + *npn, beg = 1, end = rootint(hi/m,r);
if (r > k) {
for (v = beg; v <= end; v++) {
if (gcd_ui(m,v) == 1 && is_square_free(v))
_pcg(lo, hi, k, m*ipow(v,r), r-1, pn, npn);
}
} else {
if (lo > m) {
UV lom = (lo/m)+!!(lo%m); /* ceildiv(lo,m) */
beg = rootint(lom, r);
if (ipow(beg,r) != lom) beg++;
}
for (v = beg; v <= end; v++)
*pnptr++ = m * ipow(v,r);
*npn += end-beg+1;
}
}
UV* powerful_numbers_range(UV* npowerful, UV lo, UV hi, UV k)
{
UV *pn, npn, i;
/* For small ranges it is faster to test each number vs generate. */
UV const single_thresh = ( (lo < 500000U) ? 30
: (lo < 400000000U) ? 160 : 600 )
* ((k <= 2) ? 1 : 4);
/* Like powerful_count, we ignore 0. */
if (lo < 1) lo = 1;
if (hi < lo) {
pn = 0;
npn = 0;
} else if (k <= 1) {
npn = hi-lo+1;
New(0, pn, npn, UV);
for (i = lo; i <= hi; i++)
pn[i-lo] = i;
} else if ((lo+single_thresh) > hi || lo > (UV_MAX-single_thresh)) {
New(0, pn, hi-lo+1, UV);
for (i = lo, npn = 0; i <= hi && i != 0; i++)
if (is_powerful(i,k))
pn[npn++] = i;
} else {
npn = powerful_count(hi,k) - ((lo <= 1) ? 0 : powerful_count(lo-1,k));
New(0, pn, npn, UV);
i = 0;
_pcg(lo, hi, k, 1, 2*k-1, pn, &i);
MPUassert(i == npn, "Number of powerful numbers generated != count");
sort_uv_array(pn, npn);
}
*npowerful = npn;
return pn;
}