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Numbers n such that A133364(n) <= 1.
0

%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