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.

Previous Showing 11-18 of 18 results.

A349125 Dirichlet inverse of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, -1, -2, 0, -3, 2, -5, 0, 0, 3, -7, 0, -11, 5, 6, 0, -13, 0, -17, 0, 10, 7, -19, 0, 0, 11, 0, 0, -23, -6, -29, 0, 14, 13, 15, 0, -31, 17, 22, 0, -37, -10, -41, 0, 0, 19, -43, 0, 0, 0, 26, 0, -47, 0, 21, 0, 34, 23, -53, 0, -59, 29, 0, 0, 33, -14, -61, 0, 38, -15, -67, 0, -71, 31, 0, 0, 35, -22, -73, 0, 0, 37, -79
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A008683, A064989, A151800, A349126, A349127.
Cf. also A055615, A346234, A349134.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, If[p == 2, -1, -NextPrime[p, -1]], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)
  • PARI
    A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A349125(n) = (moebius(n)*A064989(n));
  • PARI
    A349125(n) = { my(f = factor(n)); prod(i=1, #f~, if(1
  • Python
    from sympy import prevprime, factorint, prod
def f(p, e):
    return 0 if e > 1 else -1 if p == 2 else -prevprime(p)
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 13 2021

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A064989(n/d) * a(d).
a(n) = A349126(n) - A064989(n).
Multiplicative with a(p^e) = 0 if e > 1, -1 if p = 2 and -prevprime(p) otherwise. - Sebastian Karlsson, Nov 13 2021
a(n) = A008683(n) * A064989(n). [Because A064989 is fully multiplicative. See "Properties" section in the Wikipedia article]

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

A349343 Dirichlet inverse of A193356, which is defined as n if n is odd, 0 if n is even.

Original entry on oeis.org

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, 0, -31, 0, 33, 0, 35, 0, -37, 0, 39, 0, -41, 0, -43, 0, 0, 0, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 0, -59, 0, -61, 0, 0, 0, 65, 0, -67, 0, 69, 0, -71, 0, -73, 0, 0, 0, 77, 0, -79, 0, 0, 0, -83, 0, 85, 0, 87, 0, -89
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Agrees with A349341 on odd numbers.
Cf. A193356, and also A328203, A349134, A349344, A349346, A349353.

Programs

  • Mathematica
    a[1]=1;a[n_]:=-DivisorSum[n,If[OddQ[n/#],n/#,0]*a@#&,#Giorgos Kalogeropoulos, Nov 15 2021 *)
f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)
  • PARI
    A349343(n) = { my(f = factor(n)); prod(i=1, #f~, if((2==f[i,1])||(f[i,2]>1), 0, -f[i,1])); };

Formula

a(2n) = 0, a(2n+1) = A349341(2n+1) for all n >= 1.
Multiplicative with a(p^e) = 0 if p=2 or e>1, otherwise a(p) = -p. - (After Sebastian Karlsson's similar formula for A349341).

A349346 Dirichlet inverse of A181988, where A181988(n) = A001511(n)*A003602(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A001511, A003602, A181988, A349347.
Cf. also A347957, A349134, A349344.

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(dA001511(n) = 1+valuation(n,2);
A003602(n) = (1+(n>>valuation(n,2)))/2;
A181988(n) = (A001511(n)*A003602(n));
v349346 = DirInverseCorrect(vector(up_to,n,A181988(n)));
A349346(n) = v349346[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A181988(n/d) * a(d).
a(n) = A349347(n) - A181988(n).

A353366 Dirichlet inverse of A110963, which is a fractalization of Kimberling's paraphrases sequence (A003602).

Original entry on oeis.org

1, -1, -1, 0, -2, 1, -1, 0, -2, 2, -2, 0, -4, 1, 3, 0, -5, 2, -3, 0, -4, 2, -2, 0, -3, 4, 1, 0, -8, -3, -1, 0, -5, 5, -1, 0, -10, 3, 5, 0, -11, 4, -6, 0, -4, 2, -2, 0, -12, 3, 3, 0, -14, -1, 4, 0, -9, 8, -8, 0, -16, 1, 14, 0, -1, 5, -9, 0, -14, 1, -5, 0, -19, 10, -4, 0, -16, -5, -3, 0, -12, 11, -11, 0, -2, 6, 10
Offset: 1

Views

Author

Antti Karttunen, Apr 18 2022

Keywords

Links

Crossrefs

Cf. A003602, A110963, A353367.
Cf. also A349134, A353368.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
A353366(n) = v353366[n];
  • Python
    from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A353366(n): return 1 if n==1 else -sum(((1+(m:=d>>(~d&d-1).bit_length())>>(m+1&-m-1).bit_length())+1)*A353366(n//d) for d in divisors(n,generator=True) if d>1) # Chai Wah Wu, Jan 04 2024

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A110963(n/d) * a(d).
a(n) = A353367(n) - A110963(n).

A353368 Dirichlet inverse of A103391, "even fractal sequence".

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -3, -1, 2, 2, -4, -3, -3, 4, 3, 0, -2, -10, -6, 1, 8, 4, -7, 3, 1, -2, -8, -1, -5, -4, -9, -1, 14, -10, 2, 17, -6, 4, 1, -1, -4, -22, -12, 1, -3, 4, -13, -1, 6, -14, -6, 11, -8, 28, 1, 1, 19, -10, -16, 3, -9, 4, -25, -1, 10, -42, -18, 25, 18, 0, -19, -17, -6, -14, -12, 5, 13, 12, -21, 3, 24
Offset: 1

Views

Author

Antti Karttunen, Apr 18 2022

Keywords

Links

Crossrefs

Cf. A003602, A103391, A353369.
Cf. also A349134, A353366.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1)));
v353368 = DirInverseCorrect(vector(up_to,n,A103391(n)));
A353368(n) = v353368[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A103391(n/d) * a(d).
a(n) = A353369(n) - A103391(n).
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