cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A073184 Number of cubefree divisors 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, 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

Views

Author

Reinhard Zumkeller, Jul 19 2002

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Haskell
    a073184 = sum . map a212793 . a027750_row
-- Reinhard Zumkeller, May 27 2012
  • Mathematica
    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 *)
  • PARI
    a(n) = {my(e = factor(n)[,2]); prod(i = 1, #e, if(e[i] == 1, 2, 3))}; \\ Amiram Eldar, Oct 08 2022

Formula

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

A073185 Sum of cubefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 13, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 28, 57, 93, 72, 98, 54, 39, 72, 56, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68, 126, 96, 144, 72
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 19 2002

Keywords

Comments

Sum of divisors of the cubefree kernel of n (see first formula).

Examples

			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) = 1 + 2 + 4 + 7 + 14 + 28 = 56.

Links

Crossrefs

Cf. A000203, A073184, A004709, A073182, A073181, A048250.

Programs

  • Haskell
    a073185 = sum . filter ((== 1) . a212793) . a027750_row
-- Reinhard Zumkeller, May 27 2012
  • Maple
    charFfree := proc(n,t) local f; for f in ifactors(n)[2] do if op(2,f) >= t then return 0 ; end if; end do: return 1 ; end proc:
A073185 := proc(n) add( d*charFfree(d,3),d =numtheory[divisors](n) ); end proc: # R. J. Mathar, Apr 12 2011
  • Mathematica
    nn = 71;f[list_, i_] := list[[i]]; a =Table[If[Max[FactorInteger[n][[All, 2]]] <= 2, n, 0], {n, 1, nn}]; b = Table[1, {nn}]; Select[Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}], # > 0 &] (* Geoffrey Critzer, Mar 22 2015 *)
f[p_, e_] := 1 + p + If[e > 1, p^2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)
  • PARI
    a(n) = {my(f=factor(n)); for (i=1, #f~, p = f[i,1]; if ((e=f[i,2]) == 1, f[i,1] = 1+p, f[i,1] = 1+p+p^2); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Feb 06 2015

Formula

a(n) = A000203(A007948(n)).
a(n) <= A073183(n).
Multiplicative with a(p) = 1+p, a(p^e) = 1 + p + p^2, e>1. - Christian G. Bower, May 18 2005
a(n) = sum(A212793(A027750(n,k)) * A027750(n,k): k=1..A000005(n)). - Reinhard Zumkeller, May 27 2012
Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(3s-3). - R. J. Mathar, Apr 12 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*Zeta(3)). - Vaclav Kotesovec, Feb 01 2019

Extensions

Incorrect comment removed by Álvar Ibeas, Feb 06 2015

A073180 Number of divisors of n which are not greater than the squarefree kernel of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 4, 2, 5, 4, 4, 2, 5, 2, 4, 2, 5, 2, 8, 2, 2, 4, 4, 4, 5, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 5, 2, 4, 4, 5, 2, 4, 4, 6, 4, 4, 2, 11, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 5, 2, 4, 4, 5, 4, 8, 2, 6, 2, 4, 2, 11, 4, 4, 4, 6, 2, 10, 4, 5, 4, 4, 4, 5, 2, 4, 5, 5, 2, 8, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 19 2002

Keywords

Examples

			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) = 6.

Links

Crossrefs

Cf. A000005, A007947, A034444, A073181, A073182, A095960.

Programs

  • Haskell
    a073180 n = length [x | x <- a027750_row n, x <= a007947 n]
-- Reinhard Zumkeller, Sep 10 2013
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, # <= Times @@ FactorInteger[n][[;;, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

Formula

a(n) >= A034444(n).
a(n) = A095960(n) + 1. - Reinhard Zumkeller, Sep 10 2013

A073183 Sum of divisors of n that are not greater than the cubefree kernel of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 36, 31, 42, 13, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 50, 42, 96, 44, 84, 78, 72, 48, 36, 57, 93, 72, 98, 54, 39, 72, 64, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68, 126, 96, 144, 72
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 19 2002

Keywords

Comments

a(n) >= A073185(n).

Examples

			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) = 1 + 2 + 4 + 7 + 8 + 14 + 28 = 64.

Links

Crossrefs

Cf. A000203, A007948, A073182, A073181.

Programs

  • Mathematica
    sdcfk[n_]:=Module[{cf=Times@@Flatten[Table[#[[1]],#[[2]]]&/@({#[[1]],If[ #[[2]]>2,2,#[[2]]]}&/@FactorInteger[n])]},Total[Select[Divisors[n],#<= cf&]]]; Array[sdcfk,80] (* Harvey P. Dale, Jul 14 2018 *)
  • PARI
    a007948(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = min(f[i, 2], 2)); factorback(f);
a(n) = sumdiv(n, d, d*(d<=a007948(n))); \\ Michel Marcus, Feb 07 2015
Showing 1-4 of 4 results.

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