A225099 Number of ways n can be represented as a sum of two distinct nontrivial prime powers (numbers of the form p^k where p is a prime number and k >= 2).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 2, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2

Offset: 0

Author

Alex Ratushnyak, Apr 27 2013

Keywords

• nonn

Comments

Nontrivial prime powers are A025475 except the first term A025475(1) = 1.

First occurrences of terms bigger than 2: see A225100.

Among first 2^32 terms 16346685 are positive (that is 0.38%).

Among first 2^34 terms 55527808 are positive (0.32%).

Examples

			12 = 2^2 + 2^3, so a(12) = 1.
36 = 32 + 4 = 27 + 9, so a(36) = 2.
		

Crossrefs

Cf. A025475, A225100.

Programs

C

#include 
#include 
#define TOP (1ULL<<17)
unsigned long long *powers, pwFlat[TOP], primes[TOP] = {2};
int main() {
  unsigned long long a, c, i, j, k, n, p, r, pp = 1, pfp = 0;
  powers = (unsigned long long*)malloc(TOP * TOP/8);
  memset(powers, 0, TOP * TOP/8);
  for (a = 3; a < TOP; a += 2) {
    for (p = 0; p < pp; ++p)  if (a % primes[p] == 0) break;
    if (p == pp)  primes[pp++] = a;
  }
  for (k = i = 0; i < pp; ++i)
    for (j = primes[i]*primes[i]; j < TOP*TOP; j *= primes[i])
      powers[j/64] | = 1ULL << (j & 63), ++k;
  if (k > TOP) exit(1);
  for (n = 0; n < TOP * TOP; ++n)
    if (powers[n/64] & (1ULL << (n & 63)))  pwFlat[pfp++] = n;
  for (n = 0; n < TOP * TOP; ++n) {
    for (c = i = 0; pwFlat[i] * 2 < n; ++i)
      r=n-pwFlat[i], c+= (powers[r/64] & (1ULL <<(r&63))) > 0;
    printf("%llu, ", c);
  }
  return 0;
}

Mathematica

nn = 100; p = Sort[Flatten[Table[Prime[n]^i, {n, PrimePi[Sqrt[nn]]}, {i, 2, Log[Prime[n], nn]}]]]; t =Sort[Select[Flatten[Table[p[[i]] + p[[j]], {i, Length[p] - 1}, {j, i + 1, Length[p]}]], # <= nn &]]; Table[Count[t, n], {n, 0, nn}] (* T. D. Noe, Apr 29 2013 *)