A295901 -id:A295901 - cp's OEIS frontend

A378226 XOR-Moebius transform of A318457, where A318457(n) = n XOR (sigma(n)-n).

1, 2, 3, 4, 5, 0, 7, 8, 15, 4, 11, 24, 13, 0, 1, 16, 17, 8, 19, 4, 27, 16, 23, 40, 27, 4, 27, 0, 29, 52, 31, 32, 39, 36, 45, 8, 37, 32, 57, 16, 41, 0, 43, 24, 5, 32, 47, 80, 63, 0, 53, 20, 53, 104, 41, 112, 63, 4, 59, 124, 61, 0, 7, 64, 91, 48, 67, 76, 75, 36, 71, 0, 73, 68, 103, 56, 83, 36, 79, 48, 111, 84, 83

Offset: 1

Antti Karttunen, Nov 26 2024

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = A318457(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of Xor-Moebius transform.

Cf. A000203, A001065, A003987, A318457, A378227, A378230 (positions of 0's), A378441 (fixed points).

A318457(n) = bitxor(n,sigma(n)-n);
A378226(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A318457(d)))); (v); }

A318503 Xor-Moebius transform of A001065, the sum of proper divisors.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 4, 5, 8, 1, 20, 1, 10, 9, 8, 1, 22, 1, 28, 11, 14, 1, 48, 7, 16, 9, 20, 1, 44, 1, 16, 15, 20, 13, 52, 1, 22, 17, 32, 1, 48, 1, 36, 45, 26, 1, 96, 9, 36, 21, 60, 1, 94, 17, 88, 23, 32, 1, 76, 1, 34, 39, 32, 19, 72, 1, 44, 27, 68, 1, 120, 1, 40, 63, 84, 19, 92, 1, 80, 37, 44, 1, 184, 23, 46, 33, 112, 1, 132

Offset: 1

Antti Karttunen, Aug 28 2018

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = sigma(n)-n for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of Xor-Moebius transform.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001065, A051953, A256739, A295901, A296203, A318501, A318502.

A318503(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, sigma(d)-d))); (v); } \\ after code in A295901.

a(n) = A318501(n) XOR A318502(n).

A324717 Xor-Moebius transform of A324716, where A324716(n) = bitxor(2A156552(n), bitand(2A156552(n), A323243(n))).

Original entry on oeis.org

0, 2, 4, 0, 8, 14, 16, 4, 4, 24, 32, 26, 64, 50, 8, 0, 128, 28, 256, 48, 20, 96, 512, 48, 8, 192, 4, 102, 1024, 26, 2048, 24, 100, 384, 24, 28, 4096, 770, 64, 96, 8192, 118, 16384, 192, 8, 1536, 32768, 104, 16, 58, 388, 384, 65536, 52, 40, 196, 256, 3074, 131072, 114, 262144, 6144, 68, 8, 200, 166, 524288, 772, 1540, 120, 1048576

Offset: 1

Antti Karttunen, Mar 15 2019

nonn

Properties of Xor-Moebius transform are explained in A295901.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A156552, A295901, A324712, A324713, A324716.

A324717(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A324716(d)))); (v); }; \\ For other code, see A324716.

A341335 For any number n with binary expansion (b_1, ..., b_k), the binary expansion of a(n), say (c_1, ..., c_k) satisfies c_m = Sum_{d | m} b_d mod 2 for m = 1..k.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 5, 4, 15, 14, 13, 12, 10, 11, 8, 9, 31, 30, 29, 28, 27, 26, 25, 24, 21, 20, 23, 22, 17, 16, 19, 18, 63, 62, 61, 60, 59, 58, 57, 56, 54, 55, 52, 53, 50, 51, 48, 49, 42, 43, 40, 41, 46, 47, 44, 45, 35, 34, 33, 32, 39, 38, 37, 36, 127, 126, 125

Offset: 0

Rémy Sigrist, Apr 25 2021

This sequence is a permutation of the nonnegative integers with inverse A341336.
This sequence operates on binary expansions in the same way as the XOR-Moebius transform described in A295901.
This sequence has only two fixed points: a(0) = 0, a(1) = 1.

			For n = 42:
- the binary expansion of 42 is (1, 0, 1, 0, 1, 0),
- the binary expansion of a(42) has 6 digits:
    - the 1st digit = 1                     mod 2 = 1,
    - the 2nd digit = 1 + 0                 mod 2 = 1,
    - the 3rd digit = 1     + 1             mod 2 = 0,
    - the 4th digit = 1 + 0     + 0         mod 2 = 1,
    - the 5th digit = 1             + 1     mod 2 = 0,
    - the 6th digit = 1 + 0 + 1         + 0 mod 2 = 0,
- so the binary expansion of a(42) is "110100",
- and a(42) = 52.

Cf. A295901, A341336.

a(n) = { my (b=binary(n), c=vector(#b)); for (m=1, #b, fordiv (m, d, c[m]=(c[m] + b[d])%2)); fromdigits(c, 2) }

a(n) < 2^k for any n < 2^k.
a(floor(n/2)) = floor(a(n)/2).
a(2^k) = 2^(k+1) - 1 for any k >= 0.

A378227 XOR-Moebius transform of A318467, where A318467(n) = 2*n XOR sigma(n).

Original entry on oeis.org

3, 4, 1, 8, 15, 6, 5, 16, 29, 14, 25, 12, 23, 6, 11, 32, 51, 30, 49, 12, 13, 22, 53, 24, 33, 14, 1, 12, 39, 126, 29, 64, 105, 70, 127, 20, 111, 70, 99, 24, 123, 58, 121, 12, 15, 38, 109, 48, 93, 30, 31, 28, 95, 22, 51, 24, 17, 14, 73, 172, 71, 6, 1, 128, 205, 114, 193, 140, 221, 102, 197, 72, 219, 142, 205, 108

Offset: 1

Antti Karttunen, Nov 26 2024

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = A318467(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of Xor-Moebius transform.

Cf. A000203, A003987, A256739, A296203, A318467.

Cf. also A083254, A378226.

A318467(n) = bitxor(2*n, sigma(n));
A378227(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A318467(d)))); (v); }

a(n) = 2*A256739(n) XOR A296203(n).

A331700 Binary XOR of squares of divisors of n.

Original entry on oeis.org

1, 5, 8, 21, 24, 40, 48, 85, 89, 120, 120, 168, 168, 240, 240, 341, 288, 317, 360, 504, 384, 408, 528, 680, 617, 520, 640, 1008, 840, 816, 960, 1365, 1072, 1440, 1248, 1197, 1368, 1224, 1360, 2040, 1680, 1920, 1848, 1560, 1864, 2640, 2208, 2728, 2385, 3021

Offset: 1

Rémy Sigrist, Jan 25 2020

			For n = 6:
- the divisors of 6 are 1, 2, 3 and 6,
- so a(6) = 1 XOR 4 XOR 9 XOR 36 = 40.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, Colored scatterplot of the first 2^18 terms (where the color is function of A007814(n))

Cf. A001157, A007814, A178910, A295901.

Table[BitXor@@(Divisors[n]^2),{n,50}] (* Harvey P. Dale, May 03 2023 *)

a(n) = my (s=0); fordiv (n, d, s=bitxor(s, d^2)); s

from functools import reduce
from operator import xor
from sympy import divisors
def A331700(n): return reduce(xor,(d**2 for d in divisors(n,generator=True))) # Chai Wah Wu, Jul 01 2022

cp's OEIS Frontend

A378226 XOR-Moebius transform of A318457, where A318457(n) = n XOR (sigma(n)-n).

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A318503 Xor-Moebius transform of A001065, the sum of proper divisors.

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A324717 Xor-Moebius transform of A324716, where A324716(n) = bitxor(2*A156552(n), bitand(2*A156552(n), A323243(n))).

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A341335 For any number n with binary expansion (b_1, ..., b_k), the binary expansion of a(n), say (c_1, ..., c_k) satisfies c_m = Sum_{d | m} b_d mod 2 for m = 1..k.

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A378227 XOR-Moebius transform of A318467, where A318467(n) = 2*n XOR sigma(n).

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A331700 Binary XOR of squares of divisors of n.

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A324717 Xor-Moebius transform of A324716, where A324716(n) = bitxor(2A156552(n), bitand(2A156552(n), A323243(n))).