A228018 Prime powers p (A025475) such that sum of proper divisors of p is also a prime power.

9, 49, 243, 961, 16129, 67092481, 17179607041, 274876858369, 4611686014132420609

Offset

1,1

Comments

Intersection of A025475(n) and A001065(A025475(n)).

Eight of the first nine terms are squares of Mersenne primes (A133049).

Links

Table of n, a(n) for n=1..9.

Example

Proper divisors of 243 are 1, 3, 9, 27, 81, their sum is 121 = 11^2, so 243 is in the sequence.

Prog

(C)
#include <stdio.h>
#include <stdlib.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;
}
U64 findElement(U64 *a, U64 start, U64 end, U64 element) {
  if (start+1==end)  return (a[start]==element);
  U64 mid = (start+end)/2;
  if (a[mid] > element)
    return findElement(a, start, mid, element);
  return findElement(a, mid, end, element);
}
int main() {
  U64 i, j, p, n=0, *pp = (U64*)malloc(TOP/2), sum;
  unsigned char *c = (unsigned char *)malloc(TOP/16);
  if (!c || !pp) exit(1);
  memset(c, 0, TOP/16);
  pp[n++] = 1;
  for (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<<60)*16.0)) break;
      }
      if (i>1)
        for (j=i*i>>1; j<TOP/2; j+=i)  c[j>>3] |= 1<<(j&7);
    }
    if ((i&(i-2))==1)  printf("%llu ", i);
  }
  printf("// %llu\n\n", n);
  qsort(pp, n, 8, compare64);
  for (i=1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0)
      for (p=i+(i==1), sum=1+p, j = p*p; ; j*=p) {
        if (findElement(pp, 0, n, sum)) printf("%llu, ", j);
        sum += j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<60)*16.0)) break;
      }
  return 0;
}
(PARI) for(n=1, 10^6, if(!isprime(n), v=factor(n); if(matsize(v)[1]==1, s=sumdiv(n, d, d)-n; if(!isprime(s), vv=factor(s); if(matsize(vv)[1]==1, print(n)))))) /* Ralf Stephan, Aug 05 2013 */

Crossrefs

Cf. A025475, A001065, A133049.

Sequence in context: A080026 A060867 A192814 * A081655 A181539 A224473

Adjacent sequences: A228015 A228016 A228017 * A228019 A228020 A228021

Keyword

nonn,more

Author

Alex Ratushnyak, Aug 02 2013

Status

approved