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

1,1

Links

Table of n, a(n) for n=1..40.

Example

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

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

Prog

(C)
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#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<TOP; j+=i)  c[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;
}

Crossrefs

Cf. A025475, A226231, A226232.

Sequence in context: A062307 A114130 A130100 * A258400 A173336 A277925

Adjacent sequences: A226227 A226228 A226229 * A226231 A226232 A226233

Keyword

nonn

Author

Alex Ratushnyak, May 31 2013

Status

approved