Florian Lang - cp's OEIS frontend

User: Florian Lang

Florian Lang has authored 2 sequences.

A333831 Dirichlet inverse of Mertens function (A002321).

1, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 4, 3, 2, 5, 2, 2, 4, 3, 7, 6, 1, 2, 8, 6, 1, 6, 5, 2, 9, 4, 8, 7, 2, 9, 13, 2, 1, 6, 10, 1, 10, 3, 7, 19, 2, 3, 16, 7, 7, 6, 8, 3, 14, 10, 14, 7, 0, 1, 31, 2, 1, 19, 8, 12, 7, 2, 6, 5, 14, 3, 35, 4, 3, 23, 9, 10, 11, 4, 24

Offset: 1

Florian Lang, Apr 07 2020

Florian Lang, Table of n, a(n) for n = 1..9998

Cf. A002321.

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sum(k=1, n, moebius(k))))} \\ Andrew Howroyd, Apr 07 2020

A309952 XOR contraction of binary representation of n.

Original entry on oeis.org

0, 1, 1, 0, 2, 3, 3, 2, 2, 3, 3, 2, 0, 1, 1, 0, 4, 5, 5, 4, 6, 7, 7, 6, 6, 7, 7, 6, 4, 5, 5, 4, 4, 5, 5, 4, 6, 7, 7, 6, 6, 7, 7, 6, 4, 5, 5, 4, 0, 1, 1, 0, 2, 3, 3, 2, 2, 3, 3, 2, 0, 1, 1, 0, 8, 9, 9, 8, 10, 11, 11, 10, 10, 11, 11, 10, 8, 9, 9, 8, 12, 13, 13

Offset: 0

Florian Lang, Aug 24 2019

To calculate a(n) write down the binary representation of n. Organize the digits in pairs and calculate the xor of these pairs. The result is a(n) in binary.

Conjecture: The index of the first occurrence of k in a is A000695(k). - Ivan N. Ianakiev, Aug 26 2019

			For n=19 we have the binary representation 10011 = 01 00 11. Calculating the xor of the pairs gives 1 0 0 which is 4 in binary and therefore a(19) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000695, A292371, A292372, A292373.

a:= n-> `if`(n=0, 0, (r-> 2*a((n-r)/4) +r*(3-r)/2)(irem(n, 4))):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 26 2019

a(n) = {my(b = Vecrev(binary(n)), nb = #b\2, val = fromdigits(Vecrev(vector(nb, i, bitxor(b[2*i-1], b[2*i]))), 2)); if (#b % 2, val += 2^nb); val;} \\ Michel Marcus, Aug 26 2019

def a(n):
    n = [int(k) for k in bin(n)[2:]]
    if len(n) % 2 != 0:
        n = [0] + n
    result = []
    for i in range(0, len(n), 2):
        result.append(n[i] ^ n[i+1]) #xor
    return int("".join([str(k) for k in result]), 2)

from itertools import zip_longest
from operator import xor
def A309952(n): return int(''.join(map(lambda x:str(xor(*x)),zip_longest((s:=tuple(int(d) for d in bin(n)[2:]))[::-2],s[-2::-2],fillvalue=0)))[::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A292371(n) + A292372(n). - Rémy Sigrist, Aug 25 2019

a(0) = 0, a(4n) = 2*a(n), a(4n+1) = 2*a(n)+1, a(4n+2) = 2*a(n)+1, a(4n+3) = 2*a(n). - Florian Lang, Aug 26 2019

cp's OEIS Frontend

User: Florian Lang

A333831 Dirichlet inverse of Mertens function (A002321).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A309952 XOR contraction of binary representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Python

Python

Formula