A282290 -id:A282290 - cp's OEIS frontend

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

Ilya Gutkovskiy, Feb 25 2017

nonn

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)

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

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

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

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]

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

A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 3, 0, 1, 1, 4, 3, 0, 2, 2, 2, 0, 3, 4, 3, 1, 2, 6, 3, 1, 0, 5, 4, 2, 2, 4, 3, 0, 2, 3, 5, 0, 1, 3, 4, 3, 2, 4, 3, 3, 4, 5, 4, 0, 2, 5, 5, 0, 4, 6, 2, 1, 1, 7, 3, 1, 2, 7, 4, 2, 4, 5, 5, 1, 3, 6, 5, 1, 3, 6, 6, 3, 4, 4, 4, 2, 4, 7, 6, 3, 1, 4, 4, 0, 4, 6, 5, 2, 2, 7, 5, 2, 1, 7, 8, 4

Offset: 0

Ilya Gutkovskiy, May 23 2017

nonn

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

			a(26) = 3 because we have [25, 1], [22, 4] and [16, 10].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Cf. A000469, A098983, A246547, A282192, A282290, A282318, A282947.

nmax = 120; CoefficientList[Series[(Sum[Boole[SquareFreeQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]) (Sum[Boole[PrimePowerQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

x='x+O('x^120); concat([0, 0, 0, 0, 0], Vec(sum(k=1, 120, (issquarefree(k) && !isprime(k))*x^k) * sum(k=1, 120, (isprimepower(k) && !isprime(k))*x^k))) \\ Indranil Ghosh, May 23 2017

G.f.: (Sum_{k>=1} x^A246547(k))*(Sum_{k>=1} x^A000469(k)).

cp's OEIS Frontend

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

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