A081707 -id:A081707 - cp's OEIS frontend

A162906 a(n) = n - A081707(n).

0, 1, 2, 3, 4, 4, 6, 7, 8, 8, 10, 9, 12, 12, 13, 15, 16, 15, 18, 17, 19, 20, 22, 20, 24, 24, 26, 25, 28, 25, 30, 31, 31, 32, 33, 31, 36, 36, 37, 36, 40, 37, 42, 41, 42, 44, 46, 43, 48, 47, 49, 49, 52, 50, 53, 52, 55, 56, 58, 52, 60, 60, 60, 63, 63, 61, 66, 65, 67, 65, 70, 65, 72, 72, 72

Offset: 1

Juri-Stepan Gerasimov, Jul 17 2009

			a(1)=1-1. a(2)=2-1. a(3)=3-1. a(4)=4-1. a(5)=5-1. a(6)=6-2.

Cf. A000027, A069345, A081707.

A081707 := proc(n) numtheory[tau](n)-numtheory[bigomega](n) ; end: A162906 := proc(n) n - A081707(n) ; end: seq(A162906(n),n=1..100) ; # R. J. Mathar, Jul 21 2009

a(n) = A000027(n) - A081707(n) = n + A001222(n) - A000005(n).

Duplicate of 9 removed by R. J. Mathar, Jul 21 2009

A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

First differs from A081707 at a(60) = 9, A081707(60) = 8.

			The a(60) = 9 factorizations are (2*2*3*5), (2*2*15), (2*3*10), (2*5*6), (2*30), (3*20), (5*12), (6*10), (60).

Cf. A000837, A001055, A001597, A007716, A007916, A045778, A052409, A052410, A162247, A281113, A281116, A303708, A303709, A303710.

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[facsr[n]],{n,100}]

Dirichlet g.f.: Product_{n in A007916} 1/(1 - n^s).

cp's OEIS Frontend

A162906 a(n) = n - A081707(n).

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