A226231 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect square in more than 1 way.

25, 125, 512, 3125, 4913, 50653, 78125, 7645373, 48828125, 69343957, 310288733, 410338673, 1220703125, 4103684801, 24820429213, 164296466333, 296709280757, 762939453125, 1772000266301, 10368641602001, 11392143817501, 19073486328125, 29724740184833, 72079590632113

Offset: 1

Author

Alex Ratushnyak, May 31 2013

Keywords

• nonn

Comments

Subsequence of A226230.

Examples

			125 = 4 + 11^2 = 25 + 10^2 = 121 + 4^2. Because there are three representations, 125 is in the sequence. Note that 100 + 5^2 is not listed, because 100 is not a prime power.
		

Crossrefs

Cf. A025475, A226230, A226232.

Programs

C

#include 
#include 
#include 
#define TOP (1ULL<<32)
typedef unsigned long long U64;
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1== *(U64*)p2) return 0;
  return (*(U64*)p1 < *(U64*)p2) ? -1 : 1;
}
int main() {
  U64 i, j, k, p, t, r, n=0, *pp = (U64*)malloc(TOP/2);
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (pp[n++] = i = 1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      for (p=i+(i==1), j = p*p; ; j*=p) {
        pp[n++] = j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<62)*4.0)) break;
      }
      if(i>1) for (j=i*i>>1; j>3]|= 1<<(j&7);
    }
  qsort(pp, n, 8, compare64);
  for (i = 0; i < n; i++)
    for (p=pp[i], j = k = 0; j < i; j++) {
      t = p - pp[j];
      r = sqrt(t);
      if (r*r==t && ++k==2) { printf("%llu, ", p); break; }
    }
  return 0;
}