A226230 Nontrivial prime powers (A025475) which are a sum of a smaller nontrivial prime power and a perfect square.

8, 9, 25, 32, 81, 125, 128, 169, 289, 343, 512, 625, 1681, 2048, 2197, 3125, 3721, 4913, 8192, 32761, 32768, 50653, 66049, 78125, 97969, 131072, 177241, 357911, 524288, 707281, 1030301, 1419857, 1442401, 1953125, 2097152, 2476099, 3031081, 3463321, 6355441, 7645373

Offset: 1

Author

Alex Ratushnyak, May 31 2013

Keywords

• nonn

Examples

			81 = 32 + 7^2, so 81 is in the sequence.
169 = 25 + 12^2, so 169 is in the sequence.
		

Crossrefs

Cf. A025475, A226231, 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, 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=0; j < i; j++) {
      t = p - pp[j];
      r = sqrt(t);
      if (r*r==t) { printf("%llu, ", p); break; }
    }
  return 0;
}

Maple

for n from 1 do
    if isA025475(n) then
        for j from 1 do
            pj := n-j^2 ;
            if pj < 0 then
                break;
            elif isA025475(pj) then
                printf("%d,\n",n);
                break ;
            end if;
        end do:
    end if:
end do: # R. J. Mathar, Jun 06 2013