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A282947 Number of ways of writing n as a sum of a perfect power and a squarefree semiprime. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A096830 A141647 A001617 * A287200 A284387 A143667

Adjacent sequences:  A282944 A282945 A282946 * A282948 A282949 A282950

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 25 2017

STATUS

approved

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Last modified October 19 17:55 EDT 2018. Contains 316376 sequences. (Running on oeis4.)