A286302 - OEIS

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A286302

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

0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Table of n, a(n) for n=1..35.

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

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]);

CROSSREFS

Cf. A001694, A133364.

Sequence in context: A230999 A180590 A289342 * A299767 A129268 A271317

Adjacent sequences: A286299 A286300 A286301 * A286303 A286304 A286305

KEYWORD

nonn

AUTHOR

Robert Israel, May 05 2017

STATUS

approved