A133364 -id:A133364 - cp's OEIS frontend

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

Robert Israel, May 05 2017

nonn

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.

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

Cf. A001694, A133364.

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]);

A196228 Number of ways of writing n as sum of a prime and a perfect power.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 2, 3, 1, 2, 1, 1, 1, 4, 2, 2, 3, 1, 4, 2, 2, 3, 1, 2, 5, 4, 2, 2, 2, 2, 3, 4, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 4, 2, 1, 2, 2, 2, 4, 3, 1, 2, 3, 3, 5, 4, 2, 2, 3, 2, 2, 5, 1, 4, 2, 3, 4, 2, 1, 5, 3, 1, 4, 4

Offset: 1

Philippe Deléham, Sep 29 2011

In this case, perfect power does not include 0.

Different from A133364. The first difference is at n=74, where a(n) = 2 but A133364(n) = 3.

			a(1) = a(2) = a(5) = a(1549) = a(1771561) = 0, see A119748.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A119748 (zero terms).

Cf. A000040, A001597, A133364.

nn = 100; pwrs = Union[{1}, Flatten[Table[Range[2, Floor[nn^(1/ex)]]^ex, {ex, 2, Floor[Log[2, nn]]}]]]; pp = Prime[Range[PrimePi[nn]]]; t = Table[0, {nn}]; Do[ t[[i[[1]]]] = i[[2]], {i, Tally[Sort[Select[Flatten[Outer[Plus, pwrs, pp]], # <= nn &]]]}]; t (* T. D. Noe, Sep 29 2011 *)

a(n) = Card_{n=i+j where i is in A000040 and j is in A001597}.

G.f.: (Sum_{k>=1} x^prime(k))*(Sum_{k = i^j, i>=1, j>=2} x^k). - Ilya Gutkovskiy, Feb 18 2017

Edited by Franklin T. Adams-Watters, Sep 29 2011

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

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A196228 Number of ways of writing n as sum of a prime and a perfect power.

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