A286302 Numbers n such that A133364(n) <= 1.

1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 16, 17, 22, 24, 25, 26, 31, 36, 58, 64, 76, 82, 120, 170, 193, 196, 214, 324, 328, 370, 412, 562, 676, 730, 10404

Offset: 1

Author

Robert Israel, May 05 2017

Keywords

• nonn

Comments

Numbers n such that there is at most one representation n = m+p with m in A001694 and p prime.

There are no more terms <= 10^7.

The only n <= 10^7 for which A133364(n) = 0 are 1, 2, and 5.

Conjecture: 10404 is the last term.

Links

• Math Overflow, Is every powerful number the sum of a powerful number and a prime?.

Crossrefs

Cf. A001694, A133364.

Programs

Maple

N:= 10^7: # to get all terms <= N
q:= proc(x,N) local p,R;
      R:= {x};
      for p in numtheory:-factorset(x) do
        R:= map(t -> seq(t*p^i,i=0..floor(log[p](N/t))), R)
      od;
      R
end proc:
Pow:= `union`(seq(q(n^2,N),n=1..isqrt(N))):
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
CPow:= Vector(N): CPow[convert(Pow,list)]:= 1:
CPrimes:= Vector(N): CPrimes[Primes]:= 1:
Conv:= SignalProcessing:-Convolution(CPow,CPrimes):
select(t -> Conv[t-1] < 1.5, [$2..N]);