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.

A206787 Sum of the odd squarefree divisors of n.

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 8, 1, 4, 6, 12, 4, 14, 8, 24, 1, 18, 4, 20, 6, 32, 12, 24, 4, 6, 14, 4, 8, 30, 24, 32, 1, 48, 18, 48, 4, 38, 20, 56, 6, 42, 32, 44, 12, 24, 24, 48, 4, 8, 6, 72, 14, 54, 4, 72, 8, 80, 30, 60, 24, 62, 32, 32, 1, 84, 48, 68, 18, 96, 48, 72, 4
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 12 2012

Keywords

Comments

a(A000079(n)) = 1; a(A057716(n)) > 1; a(A065119(n)) = 4; a(A086761(n)) = 6.
Inverse Mobius transform of 1, 0, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 21, 0, 23, 0, 0, 0, 0, 0, 29... - R. J. Mathar, Jul 12 2012

Links

Crossrefs

Cf. A000203, A000593, A007947, A008683, A056911, A048250, A010684, A192066, A204455.
Cf. A000079, A057716, A065119, A086761.
Inverse Möbius transform of the absolute values of A349343.

Programs

  • Haskell
    a206787 = sum . filter odd . a206778_row
  • Magma
    [&+[d:d in Divisors(m)|IsOdd(d) and IsSquarefree(d)]:m in [1..72]]; // Marius A. Burtea, Aug 14 2019
  • Maple
    seq(add(d*mobius(2*d)^2, d in divisors(n)), n=1 .. 80); # Ridouane Oudra, Aug 14 2019
  • Mathematica
    a[n_] := DivisorSum[n, #*Boole[OddQ[#] && SquareFreeQ[#]]&]; Array[a, 80] (* Jean-François Alcover, Dec 05 2015 *)
f[2, e_] := 1; f[p_, e_] := p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)
  • PARI
    a(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ Michel Marcus, Sep 21 2014
  • Python
    from math import prod
from sympy import primefactors
def A206787(n): return prod(1+(p if p>2 else 0) for p in primefactors(n)) # Chai Wah Wu, Oct 10 2024

Formula

a(n) = Sum_{k = 1..A034444(n)} (A206778(n,k) mod 2) * A206778(n,k).
a(n) = Sum_{d|n} d*mu(2*d)^2, where mu is the Möbius function (A008683). - Ridouane Oudra, Aug 14 2019
Multiplicative with a(2^e) = 1, and a(p^e) = p + 1 for p > 2. - Amiram Eldar, Sep 18 2020
Sum_{k=1..n} a(k) ~ (1/3) * n^2. - Amiram Eldar, Nov 17 2022
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-2))*(2^s/(2^s+2)). - Amiram Eldar, Jan 03 2023
From Antti Karttunen, Nov 22 2023: (Start)
a(n) = A000203(A204455(n)) = A000593(A007947(n)) = A048250(n)/A010684(n-1). [From Sequence Machine]
a(n) = Sum_{d|n} abs(A349343(d)). [See R. J. Mathar's Jul 12 2012 comment above]
(End)
a(n) = Sum_{d divides n, d odd} d * mu(d)^2. - Peter Bala, Feb 01 2024

A349344 Dirichlet inverse of A109168, where A109168(n) = (n+A006519(n))/2, and A006519 is the highest power of 2 dividing n.

Original entry on oeis.org

1, -2, -2, 0, -3, 4, -4, 0, -1, 6, -6, 0, -7, 8, 4, 0, -9, 2, -10, 0, 5, 12, -12, 0, -4, 14, -2, 0, -15, -8, -16, 0, 7, 18, 6, 0, -19, 20, 8, 0, -21, -10, -22, 0, 3, 24, -24, 0, -9, 8, 10, 0, -27, 4, 8, 0, 11, 30, -30, 0, -31, 32, 4, 0, 9, -14, -34, 0, 13, -12, -36, 0, -37, 38, 8, 0, 9, -16, -40, 0, -4, 42, -42, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A109168 (A140472), A349345.
Cf. also A328203, A349134, A349343, A349346, A349353.

Programs

  • PARI
    up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA109168(n) = ((n+bitand(n, -n))\2); \\ From A109168 by M. F. Hasler, Oct 19 2019 (Cf. A140472).
v349344 = DirInverseCorrect(vector(up_to,n,A109168(n)));
A349344(n) = v349344[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A109168(n/d) * a(d).
a(n) = A349345(n) - A109168(n).

A349353 Dirichlet inverse of A328203.

Original entry on oeis.org

1, -2, -5, 0, -8, 10, -11, 0, 5, 16, -17, 0, -20, 22, 38, 0, -26, -10, -29, 0, 52, 34, -35, 0, 11, 40, 1, 0, -44, -76, -47, 0, 80, 52, 82, 0, -56, 58, 94, 0, -62, -104, -65, 0, -34, 70, -71, 0, 19, -22, 122, 0, -80, -2, 126, 0, 136, 88, -89, 0, -92, 94, -46, 0, 148, -160, -101, 0, 164, -164, -107, 0, -110, 112, -45
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Dirichlet convolution of A349134 with A349341, or equally of A349343 with A349344.
Cf. A328203, A349354.

Programs

  • PARI
    up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2));
v349353 = DirInverseCorrect(vector(up_to,n,A328203(n)));
A349353(n) = v349353[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A328203(n/d) * a(d).
a(n) = A349354(n) - A328203(n).
a(n) = Sum_{d|n} A349134(d) * A349341(n/d).
a(n) = Sum_{d|n} A349343(d) * A349344(n/d).

A349341 Dirichlet inverse of A026741, which is defined as n if n is odd, n/2 if n is even.

Original entry on oeis.org

1, -1, -3, -1, -5, 3, -7, -1, 0, 5, -11, 3, -13, 7, 15, -1, -17, 0, -19, 5, 21, 11, -23, 3, 0, 13, 0, 7, -29, -15, -31, -1, 33, 17, 35, 0, -37, 19, 39, 5, -41, -21, -43, 11, 0, 23, -47, 3, 0, 0, 51, 13, -53, 0, 55, 7, 57, 29, -59, -15, -61, 31, 0, -1, 65, -33, -67, 17, 69, -35, -71, 0, -73, 37, 0, 19, 77, -39, -79
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Agrees with A349343 on odd numbers.
Cf. A026741, A349342.
Cf. also A328203, A349134, A349344, A349346, A349353.

Programs

  • Mathematica
    a[1]=1;a[n_]:=-DivisorSum[n,If[OddQ[n/#],n/#,n/(2#)]*a@#&,#Giorgos Kalogeropoulos, Nov 15 2021 *)
f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)
  • PARI
    A349341(n) = { my(f = factor(n)); prod(i=1, #f~, if(2==f[i,1], -1, if(1==f[i,2], -f[i,1], 0))); };
  • Python
    from sympy import prevprime, factorint, prod
def f(p, e):
    return -1 if p == 2 else 0 if e > 1 else -p
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 15 2021

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A026741(n/d) * a(d).
a(n) = A349342(n) - A026741(n).
a(2n+1) = A349343(2n+1) for all n >= 1.
Multiplicative with a(2^e) = -1, a(p) = -p and a(p^e) = 0 if e > 1. - Sebastian Karlsson, Nov 15 2021
Showing 1-4 of 4 results.

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