A282947 Number of ways of writing n as a sum of a perfect power and a squarefree semiprime.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 1, 0, 2, 2, 0, 0, 3, 3, 1, 1, 2, 1, 0, 1, 4, 3, 0, 1, 2, 3, 1, 3, 3, 3, 2, 2, 7, 3, 1, 0, 4, 5, 2, 2, 3, 3, 1, 2, 3, 4, 1, 1, 4, 5, 3, 2, 4, 4, 3, 3, 6, 3, 0, 2, 6, 6, 0, 4, 4, 3, 1, 1, 7, 1, 1, 2, 5, 5, 2, 4, 4, 6, 2, 3, 6, 4, 2, 3, 6, 6, 4, 3, 4, 4, 2, 5, 6, 5, 3, 1, 3, 5, 0, 3, 6, 3, 3, 2, 6, 5, 3, 1, 5, 7, 5

Offset

0,15

Comments

Conjecture: a(n) > 0 for all n > 108.

From Robert G. Wilson v, Feb 25 2017: (Start)

Conjecture: a(n) > 1 for all n > 604,

Conjecture: a(n) > 2 for all n > 1008, etc.

First occurrence of k: 0, 7, 14, 22, 30, 47, 66, 42, 127, 138, 150, 222, 251, 303, 210, 430, 330, 462, 670, 770, 983, 878, 1038, 1142, 1355, 1482, ... (End)

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Ilya Gutkovskiy, Extended graphical example

Eric Weisstein's World of Mathematics, Perfect Powers

Eric Weisstein's World of Mathematics, Semiprime

Eric Weisstein's World of Mathematics, Squarefree

Formula

G.f.: (Sum_{k>=1} x^A001597(k))*(Sum_{k>=1} x^A006881(k)).

Example

a(22) = 3 because we have [21, 1], [16, 6] and [14, 8].

Mathematica

nmax = 120; CoefficientList[Series[(x + Sum[Boole[GCD @@ FactorInteger[k][[All, 2]] > 1] x^k, {k, 2, nmax}]) (Sum[MoebiusMu[k]^2 Floor[2/PrimeOmega[k]] Floor[PrimeOmega[k]/2] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Crossrefs

Cf. A001597, A006881, A196228, A282192, A282290.

Sequence in context: A001617 A333628 A342707 * A287200 A284387 A143667

Adjacent sequences: A282944 A282945 A282946 * A282948 A282949 A282950

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 25 2017

Status

approved