A073180 -id:A073180 - cp's OEIS frontend

A073184 Number of cubefree divisors of n.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4, 6, 2, 12, 4, 6, 4, 4, 4, 6, 2, 6, 6, 9, 2, 8, 2

Offset: 1

Reinhard Zumkeller, Jul 19 2002

a(n) = number of divisors of the cubefree kernel of n: a(n) = A000005(A007948(n)); [corrected by Amiram Eldar, Oct 08 2022]

Multiplicative because it is the Inverse Möbius transform of the characteristic function of cubefree numbers. a(n) is a prime signature sequence. a(p) = 2, a(p^e) = 3, e>1. - Christian G. Bower, May 18 2005

			The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, 8=2^3 and 56=7*2^3 are not cubefree, therefore a(56) = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000012, A001620, A004709, A027750, A034444, A073180, A073183, A073185, A212793.

a073184 = sum . map a212793 . a027750_row
-- Reinhard Zumkeller, May 27 2012

a[1] = 1; a[p_?PrimeQ] = 2; a[n_] := Times @@ (If[#[[2]] == 1, 2, 3] & /@ FactorInteger[n]); Table[a[n], {n, 1, 103}] (* Jean-François Alcover, May 24 2012, after Christian G. Bower *)

a(n) = {my(e = factor(n)[,2]); prod(i = 1, #e, if(e[i] == 1, 2, 3))}; \\ Amiram Eldar, Oct 08 2022

a(n) <= A073182(n).
Dirichlet g.f.: zeta(s)^2/zeta(3*s). Dirichlet convolution of the characteristic function of cubefree numbers by A000012. - R. J. Mathar, Apr 12 2011
a(n) = Sum_{k = 1..A000005(n)} A212793(A027750(n,k)). - Reinhard Zumkeller, May 27 2012
Sum_{k=1..n} a(k) ~ n / zeta(3) * (log(n) - 1 + 2*gamma - 3*zeta'(3)/zeta(3)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019

A073182 Number of divisors of n which are not greater than the cubefree kernel of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 7, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 7, 2, 8, 2, 6, 6, 4, 2, 7, 3, 6, 4, 6, 2, 6, 4, 7, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 11, 2, 4, 6, 6, 4, 8, 2, 8, 3, 4, 2, 12, 4, 4, 4, 7, 2, 12, 4, 6, 4, 4, 4, 7, 2, 6, 6, 9, 2, 8

Offset: 1

Reinhard Zumkeller, Jul 19 2002

nonn

a(n) >= A073184(n).

			The cubefree kernel of 56 = 7*2^3 is 28 = 7*2^2 and the divisors <= 28 of 56 are {1, 2, 4, 7, 8, 14, 28}, therefore a(56) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A007948, A073183, A073180.

Table[Function[k, DivisorSum[n, 1 &, # <= k &]]@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> p^Min[e, 2]], {n, 102}] (* Michael De Vlieger, Jul 18 2017 *)

a007948(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = min(f[i, 2], 2)); factorback(f);
a(n) = my(cfk = a007948(n)); sumdiv(n, d, d<=cfk); \\ Michel Marcus, May 14 2015

A073181 Sum of divisors of n which are not greater than the squarefree kernel of n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 16, 14, 24, 24, 3, 18, 12, 20, 22, 32, 36, 24, 16, 6, 42, 4, 28, 30, 72, 32, 3, 48, 54, 48, 16, 38, 60, 56, 30, 42, 96, 44, 40, 33, 72, 48, 16, 8, 18, 72, 46, 54, 12, 72, 36, 80, 90, 60, 108, 62, 96, 41, 3, 84, 144, 68, 58, 96, 144, 72, 16, 74

Offset: 1

Reinhard Zumkeller, Jul 19 2002

			The squarefree kernel of 56 = 7 * 2^3 is 14 = 7*2 and the divisors <= 14 of 56 are {1, 2, 4, 7, 8, 14}, therefore a(56) = 1 + 2 + 4 + 7 + 8 + 14 = 36.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A007947, A048250, A073180, A073183.

a[n_] := DivisorSum[n, # &, # <= Times @@ FactorInteger[n][[;;, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ This function from Andrew Lelechenko
A073181(n) = sumdiv(n, d, d*(d<=A007947(n))); \\ Antti Karttunen, Sep 10 2017, after Michel Marcus's code for A073183.

a(n) >= A048250(n).

A095960 Number of divisors of n that are less than the squarefree kernel of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 3, 3, 4, 1, 3, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3, 5, 1, 9, 3, 4, 3, 3, 3, 4, 1, 3, 4, 4, 1, 7

Offset: 1

Reinhard Zumkeller, Jul 14 2004

nonn

			n = 40 = 5*2^3, rad(40) = 10: a(40) = #{d:d<10} = #{1,2,4,5,8} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A005117, A007947, A027750, A032741, A073180.

a095960 n = length [x | x <- a027750_row n, x < a007947 n]
-- Reinhard Zumkeller, Sep 10 2013

a[n_] := DivisorSum[n, 1 &, # < Times @@ FactorInteger[n][[;;, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

a(A005117(n)) = A032741(A005117(n)).

a(A000040(n)) = 1.

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A073184 Number of cubefree divisors of n.

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A073182 Number of divisors of n which are not greater than the cubefree kernel of n.

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A073181 Sum of divisors of n which are not greater than the squarefree kernel of n.

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PARI

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A095960 Number of divisors of n that are less than the squarefree kernel of n.

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