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%I #6 May 06 2017 10:23:48
%S 1,2,3,4,5,7,8,9,10,13,16,17,22,24,25,26,31,36,58,64,76,82,120,170,
%T 193,196,214,324,328,370,412,562,676,730,10404
%N Numbers n such that A133364(n) <= 1.
%C Numbers n such that there is at most one representation n = m+p with m in A001694 and p prime.
%C There are no more terms <= 10^7.
%C The only n <= 10^7 for which A133364(n) = 0 are 1, 2, and 5.
%C Conjecture: 10404 is the last term.
%H Math Overflow, <a href="https://mathoverflow.net/questions/269080/is-every-powerful-number-the-sum-of-a-powerful-number-and-a-prime#comment665409_269080">Is every powerful number the sum of a powerful number and a prime?</a>.
%p N:= 10^7: # to get all terms <= N
%p q:= proc(x,N) local p,R;
%p R:= {x};
%p for p in numtheory:-factorset(x) do
%p R:= map(t -> seq(t*p^i,i=0..floor(log[p](N/t))), R)
%p od;
%p R
%p end proc:
%p Pow:= `union`(seq(q(n^2,N),n=1..isqrt(N))):
%p Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
%p CPow:= Vector(N): CPow[convert(Pow,list)]:= 1:
%p CPrimes:= Vector(N): CPrimes[Primes]:= 1:
%p Conv:= SignalProcessing:-Convolution(CPow,CPrimes):
%p select(t -> Conv[t-1] < 1.5, [$2..N]);
%Y Cf. A001694, A133364.
%K nonn
%O 1,2
%A _Robert Israel_, May 05 2017
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