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

Author

Ilya Gutkovskiy, Feb 25 2017

Keywords

• nonn

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)

Examples

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

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

Crossrefs

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

Programs

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]

Formula

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