A178910 -id:A178910 - cp's OEIS frontend

A226643 Numbers n such that the binary XOR of the divisors of n (A178910) is a binary palindrome (A006995) and not a power of 2 (A000079).

81, 162, 169, 324, 338, 648, 676, 1296, 1352, 2401, 2592, 2704, 3249, 4802, 5184, 5408, 6498, 9604, 10368, 10816, 12996, 19208, 20736, 21632, 25992, 38416, 41472, 43264, 51984, 76832, 82944, 86528, 103968, 112225, 153664, 165888, 173056, 194481

Offset: 1

Robert G. Wilson v, Aug 18 2013

A takeoff of A227843.

Cf. A000079, A006995, A178910, A227843.

f[n_] := Fold[ BitXor[#1, #2] &, 0, Divisors@ n]; palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse@ idn]; fQ[n_] := palQ[ f@ n, 2] && ! IntegerQ@ Log2@ n; Select[ Range@ 200000, fQ]

A028982 Squares and twice squares.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 72, 81, 98, 100, 121, 128, 144, 162, 169, 196, 200, 225, 242, 256, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 512, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1024

Offset: 1

Patrick De Geest

Numbers n such that sum of divisors of n (A000203) is odd.
Also the numbers with an odd number of run sums (trapezoidal arrangements, number of ways of being written as the difference of two triangular numbers). - Ron Knott, Jan 27 2003
Pell(n)*Sum_{k|n} 1/Pell(k) is odd, where Pell(n) is A000129(n). - Paul Barry, Oct 12 2005
Number of odd divisors of n (A001227) is odd. - Vladeta Jovovic, Aug 28 2007
A071324(a(n)) is odd. - Reinhard Zumkeller, Jul 03 2008
Sigma(a(n)) = A000203(a(n)) = A152677(n). - Jaroslav Krizek, Oct 06 2009
Numbers n such that sum of odd divisors of n (A000593) is odd. - Omar E. Pol, Jul 05 2016
A187793(a(n)) is odd. - Timothy L. Tiffin, Jul 18 2016
If k is odd (k = 2m+1 for m >= 0), then 2^k = 2^(2m+1) = 2*(2^m)^2.  If k is even (k = 2m for m >= 0), then 2^k = 2^(2m) = (2^m)^2.  So, the powers of 2 sequence (A000079) is a subsequence of this one. - Timothy L. Tiffin, Jul 18 2016
Numbers n such that A175317(n) = Sum_{d|n} pod(d) is odd, where pod(m) = the product of divisors of m (A007955). - Jaroslav Krizek, Dec 28 2016
Positions of zeros in A292377 and A292383, positions of ones in A286357 and A292583. (See A292583 for why.) - Antti Karttunen, Sep 25 2017
Numbers of the form A000079(i)*A016754(j), i,j>=0. - R. J. Mathar, May 30 2020
Equivalently, numbers whose odd part is square. Cf. A042968. - Peter Munn, Jul 14 2020
These are the Heinz numbers of the partitions counted by A119620. - Gus Wiseman, Oct 29 2021
Numbers m whose abundance, A033880(m), is odd. - Peter Munn, May 23 2022
Numbers with an odd number of middle divisors (cf. A067742). - Omar E. Pol, Aug 02 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 5.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, Journal of Integer Sequences, Vol. 16 (2013), #13.1.8.
Patrick De Geest, World!Of Numbers
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163.
Eric Weisstein's World of Mathematics, Abundance

Complement of A028983.
Characteristic function is A053866, A093709.
Odd terms in A178910.
Cf. A000203, A000290, A000593, A001105, A042968, A187793.
Supersequence of A000079.
Cf. A028260, A033880, A046951, A067742.
Cf. A119620, A119899, A347437, A347438, A348550.

import Data.List.Ordered (union)
a028982 n = a028982_list !! (n-1)
a028982_list = tail $ union a000290_list a001105_list
-- Reinhard Zumkeller, Jun 27 2015

Take[ Sort[ Flatten[ Table[{n^2, 2n^2}, {n, 35}] ]], 57] (* Robert G. Wilson v, Aug 27 2004 *)

list(lim)=vecsort(concat(vector(sqrtint(lim\1),i,i^2), vector(sqrtint(lim\2),i,2*i^2))) \\ Charles R Greathouse IV, Jun 16 2011

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A028982_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:int(is_square(n) or is_square(n<<1)),count(max(startvalue,1)))
A028982_list = list(islice(A028982_gen(),30)) # Chai Wah Wu, Jan 09 2023

from math import isqrt
def A028982(n):
    def f(x): return n-1+x-isqrt(x)-isqrt(x>>1)
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 22 2024

A001105 UNION A000290.
a(n) is asymptotic to c*n^2 with c = 2/(1+sqrt(2))^2 = 0.3431457.... - Benoit Cloitre, Sep 17 2002
In particular, a(n) = c*n^2 + O(n). - Charles R Greathouse IV, Jan 11 2013
a(A003152(n)) = n^2; a(A003151(n)) = 2*n^2. - Enrique Pérez Herrero, Oct 09 2013
Sum_{n>=1} 1/a(n) = Pi^2/4. - Amiram Eldar, Jun 28 2020

A028983 Numbers whose sum of divisors is even.

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82

Offset: 1

Patrick De Geest

The even terms of this sequence are the even terms appearing in A178910. [Edited by M. F. Hasler, Oct 02 2014]
A071324(a(n)) is even. - Reinhard Zumkeller, Jul 03 2008
Sigma(a(n)) = A000203(a(n)) = A152678(n). - Jaroslav Krizek, Oct 06 2009
A083207 is a subsequence. - Reinhard Zumkeller, Jul 19 2010
Numbers k such that the number of odd divisors of k (A001227) is even. - Omar E. Pol, Apr 04 2016
Numbers k such that the sum of odd divisors of k (A000593) is even. - Omar E. Pol, Jul 05 2016
Numbers with a squarefree part greater than 2. - Peter Munn, Apr 26 2020
Equivalently, numbers whose odd part is nonsquare. Compare with the numbers whose square part is even (i.e., nonodd): these are the positive multiples of 4, A008586\{0}, and A225546 provides a self-inverse bijection between the two sets. - Peter Munn, Jul 19 2020
Also numbers whose reversed prime indices have alternating product > 1, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). Also Heinz numbers of the partitions counted by A347448. - Gus Wiseman, Oct 29 2021
Numbers whose number of middle divisors is not odd (cf. A067742). - Omar E. Pol, Aug 02 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

The complement is A028982 = A000290 U A001105.
Cf. A000203, A000593, A001227, A007913, A178910, A152678, A067742.
Subsequences: A083207, A091067, A145204\{0}, A225838, A225858.
Cf. A334748 (a permutation).
Related to A008586 via A225546.
Ranks the partitions counted by A347448, complement A119620.
Cf. A030059, A335433, A335448, A339890, A344607, A347438, A347443, A347445, A347446, A347452, A347453, A347465.

Select[Range[82],EvenQ[DivisorSigma[1,#]]&] (* Jayanta Basu, Jun 05 2013 *)

is(n)=!issquare(n)&&!issquare(n/2) \\ Charles R Greathouse IV, Jan 11 2013

from math import isqrt
def A028983(n):
    def f(x): return n-1+isqrt(x)+isqrt(x>>1)
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 22 2024

a(n) ~ n. - Charles R Greathouse IV, Jan 11 2013
a(n) = n + (1 + sqrt(2)/2)*sqrt(n) + O(1). - Charles R Greathouse IV, Sep 01 2015
A007913(a(n)) > 2. - Peter Munn, May 05 2020

A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

Original entry on oeis.org

1, 5, 8, 20, 24, 40, 48, 80, 88, 120, 120, 160, 168, 240, 240, 320, 288, 312, 360, 480, 384, 408, 528, 640, 616, 520, 648, 960, 840, 816, 960, 1280, 1072, 1440, 1248, 1248, 1368, 1224, 1360, 1920, 1680, 1920, 1848, 1632, 1872, 2640, 2208, 2560, 2384, 3016

Offset: 1

Rémy Sigrist, Nov 29 2017

This sequence is a variant of A256739; both sequences have nice graphical features.
Replacing "SumXOR" by "Sum" in the name leads to the Jordan function J_2 (A007434).
For any sequence f of nonnegative integers with positive indices:
- let x_f be the unique sequence satisfying SumXOR_{d divides n} x_f(d) = f(n) for any n > 0,
- in particular, x_A000027 = A256739 and x_A000290 = a (this sequence),
- also, x_A178910 = A000027 and x_A055895 = A000079,
- see the links section for a gallery of x_f plots for some classic f functions,
- x_f(1) = f(1),
- x_f(p) = f(1) XOR f(p) for any prime p,
- x_A063524 = A008966,
- x_f(n) = SumXOR_{d divides n and n/d is squarefree} f(d) for any n > 0,
- the function x: f -> x_f is a bijection,
- A000004 is the only fixed point of x (i.e. x_f = f if and only if f = A000004),
- for any sequence f, x_{2*f} = 2 * x_f,
- for any sequences g and f, x_{g XOR f} = x_g XOR x_f.
From Antti Karttunen, Dec 29 2017: (Start)
The transform x_f described above could be called "Xor-Moebius transform of f" because of its analogous construction to Möbius transform with A008683 replaced by A008966 and the summation replaced by cumulative XOR.
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..16383
Rémy Sigrist, Colored scatterplot of the first 2^17-1 terms (where the color is function of A087207(n) % 8)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000010 (Euler totient function)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000203 (sigma)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A001157 (sigma_2)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000040 (prime numbers)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000720 (PrimePi)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A006370 (Collatz map)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A005132 (Recamán's sequence)

Cf. A000004, A000010, A000027, A000040, A000079, A000290, A000720, A001157, A003987, A005132, A006370, A007434, A008966, A055895, A063524, A087207, A178910.

Same transform applied to other sequences: A256739 (A000027), A296206 (A256739), A296207 (A227320), A296203 (A000203), A296208 (A005187), A297110 (A006068), A297108 (A248663), A297106 (A156552).

a(n{, f=k->k^2}) = my (v=0); fordiv(n,d,if (issquarefree(n/d), v=bitxor(v,f(d)))); return (v)

a(n) = SumXOR_{d divides n and n/d is squarefree} d^2.

A072594 In prime factorization of n replace multiplication with bitwise logical 'xor'.

Original entry on oeis.org

1, 2, 3, 0, 5, 1, 7, 2, 0, 7, 11, 3, 13, 5, 6, 0, 17, 2, 19, 5, 4, 9, 23, 1, 0, 15, 3, 7, 29, 4, 31, 2, 8, 19, 2, 0, 37, 17, 14, 7, 41, 6, 43, 11, 5, 21, 47, 3, 0, 2, 18, 13, 53, 1, 14, 5, 16, 31, 59, 6, 61, 29, 7, 0, 8, 10, 67, 17, 20, 0, 71, 2, 73, 39, 3, 19, 12, 12, 79, 5, 0, 43, 83, 4

Offset: 1

Reinhard Zumkeller, Jun 23 2002

n is prime iff a(n)=n;
for primes p, k>0: a(p^k)=p*(k mod 2);
for m>1: a(m^2)=0, see A072595.
a(A127812(n)) = n and a(m) <> n for m < A127812(n).

			a(35) = a(5*7) = a(5) 'xor' a(7) = '101' xor '111' = '010' = 2.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072591, A072593, A027746, A178910.

import Data.Bits (xor)
a072594 = foldl1 xor . a027746_row :: Integer -> Integer
-- Reinhard Zumkeller, Nov 17 2012

a[n_] := BitXor @@ Flatten[ Table[ First[#], {Last[#]} ]& /@ FactorInteger[n] ]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Mar 11 2013 *)

a(n)=if(n==1, return(1)); my(f=factor(n),t); for(i=1,#f~, if(f[i,2]%2, t=bitxor(t,f[i,1]))); t \\ Charles R Greathouse IV, Aug 28 2016

from sympy import factorint
from operator import _xor_
from functools import reduce
def a(n): return reduce(_xor_, (f for f in factorint(n, multiple=True))) if n > 1 else 1
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, May 31 2025

A169813 a(n) = n XOR sigma(n), where sigma(n) is the number of divisors of n, A000203.

Original entry on oeis.org

0, 1, 7, 3, 3, 10, 15, 7, 4, 24, 7, 16, 3, 22, 23, 15, 3, 53, 7, 62, 53, 50, 15, 36, 6, 48, 51, 36, 3, 86, 63, 31, 17, 20, 19, 127, 3, 26, 31, 114, 3, 74, 7, 120, 99, 102, 31, 76, 8, 111, 123, 86, 3, 78, 127, 64, 105, 96, 7, 148, 3, 94, 87, 63, 21, 210, 7, 58, 37, 214, 15, 139, 3, 56

Offset: 1

N. J. A. Sloane, May 28 2010

Cf. A000203, A003987, A070883, A169810, A169811, A169812, A169814.

Cf. also A106409, A178910, A227320, A294899.

a(n) = bitxor(n, sigma(n)); \\ Michel Marcus, Nov 25 2017

(define (A169813 n) (A003987bi n (A000203 n))) ;; Where A003987bi implements the bitwise-XOR, A003987 and code for A000203 can be found under that entry. - Antti Karttunen, Nov 25 2017

A227320 Binary XOR of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 2, 6, 1, 2, 1, 4, 7, 15, 1, 15, 1, 8, 5, 8, 1, 6, 4, 14, 11, 14, 1, 6, 1, 31, 9, 18, 3, 21, 1, 16, 15, 20, 1, 26, 1, 26, 1, 20, 1, 14, 6, 21, 19, 16, 1, 6, 15, 26, 17, 30, 1, 4, 1, 28, 25, 63, 9, 58, 1, 52, 21, 38, 1, 33, 1, 38, 17, 50, 13, 54, 1

Offset: 1

Alex Ratushnyak, Jul 06 2013

An alternative definition (with A027751) would define a(1)=1. - R. J. Mathar, Jul 14 2013

However, this definition is more aligned with A001065 and A218403 where the initial term a(1) is also 0. - Antti Karttunen, Oct 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A001065, A027751, A178910, A218403, A248663, A293214, A293215.

Array[BitXor @@ Most@ Divisors@ # &, 79] (* Michael De Vlieger, Oct 08 2017 *)

A227320(n) = { my(s=0); fordiv(n,d,if(dAntti Karttunen, Oct 08 2017

a(n) = A178910(n) XOR n, where XOR is the binary logical exclusive or operator.
From Antti Karttunen, Oct 08 2017: (Start)
a(n) = A248663(A293214(n)).
a(n) <= A218403(n) <= A001065(n).
(End)

A178908 GF(2) sum of divisors of n.

Original entry on oeis.org

1, 3, 2, 7, 7, 6, 6, 15, 12, 9, 10, 14, 12, 10, 8, 31, 25, 20, 18, 21, 19, 30, 24, 30, 24, 20, 18, 18, 20, 24, 30, 63, 60, 43, 40, 36, 36, 54, 54, 45, 40, 53, 48, 54, 48, 40, 46, 62, 60, 40, 42, 36, 36, 54, 54, 34, 36, 60, 58, 56, 60, 34, 38, 127, 121, 68, 66, 79, 79, 120, 120

Offset: 1

Franklin T. Adams-Watters, Jun 22 2010

nonn

Take the n-th GF(2) polynomial, compute its sum of divisors, and find the index of that polynomial in the list of GF(2) polynomials.
If 2^k <= n < 2^(k+1), then also 2^k <= a(n) < 2^(k+1), since any proper divisor of a GF(2) polynomial has lower degree.
Numbers whose binary representations correspond to the divisors occur as the nonzero terms on row n of A280499, and they are XORed together to obtain a(n). A280493 gives another GF(2)[X]-analog of A000203. - Antti Karttunen, Jan 11 2017

			5 => x^2 + 1 = (x+1)^2. sigma((x+1)^2) = (x+1)^2 + x+1 + 1 = x^2 + x + 1 => 7, so a(5) = 7. (All polynomials here are over GF(2).)

Antti Karttunen, Table of n, a(n) for n = 1..2047
Index entries for sequences operating on polynomials in ring GF(2)[X]

Cf. A000203, A000225, A003987, A091220, A091255, A178909, A178910, A280493, A280499.

a(n)={local(p,fm,r,k);
while(n>0,p+=Mod(n,2)*x^k;n\=2;k++);
r=Mod(1,2);fm=factor(p);for(k=1,matsize(fm)[1],r*=(fm[k,1]^(fm[k,2]+1)-1)/(fm[k,1]-1));
subst(lift(r),x,2)}

a(n) = {my(s = vecsum(divisors(Mod(1,2)*Pol(binary(n))))); subst(lift(s), x, 2);} \\ Michel Marcus, Jan 13 2019

;; A003987bi implements the 2-argument bitwise-XOR function (A003987).
;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255) which is used for checking that k is a divisor of n.
(define (A178908 n) (let loop ((k n) (s 0)) (if (zero? k) s (loop (- k 1) (A003987bi s (if (= k (A091255bi n k)) k 0))))))
;; Antti Karttunen, Jan 11 2017

For all n >= 0, a(2^n) = A000203(2^n) = A280493(2^n) = A000225(1+n). - Antti Karttunen, Jan 11 2017

A296207 Xor-Moebius transform of A227320, binary XOR of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 4, 3, 6, 1, 0, 1, 4, 7, 8, 1, 12, 1, 12, 5, 8, 1, 0, 5, 14, 9, 8, 1, 6, 1, 16, 9, 18, 3, 24, 1, 16, 15, 24, 1, 26, 1, 16, 5, 20, 1, 0, 7, 22, 19, 28, 1, 0, 15, 16, 17, 30, 1, 12, 1, 28, 31, 32, 9, 58, 1, 36, 21, 38, 1, 48, 1, 38, 19, 32, 13, 54, 1, 48, 27, 42, 1, 52, 21, 40, 31, 32, 1, 34, 11, 40, 29, 44, 23, 0, 1

Offset: 1

Antti Karttunen, Dec 25 2017

Unique sequence satisfying SumXOR_{d divides n} a(d) = A227320(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..16384
Index entries for sequences related to binary expansion of n

Cf. A003987, A178910, A227320, A256739, A295901, A296206.

Cf. A297107 (positions of zeros).

A227320(n) = { my(s=0); fordiv(n,d,if(dA296207(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A227320(d)))); (v); } \\ after code in A295901.

a(n) = n XOR A256739(n), where XOR is the bitwise XOR-operation (A003987).

cp's OEIS Frontend

A227843 Numbers n such that the binary XOR of the divisors of n (A178910(n)) is a binary repunit (A000225).

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A226643 Numbers n such that the binary XOR of the divisors of n (A178910) is a binary palindrome (A006995) and not a power of 2 (A000079).

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A028982 Squares and twice squares.

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A028983 Numbers whose sum of divisors is even.

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A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

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A072594 In prime factorization of n replace multiplication with bitwise logical 'xor'.

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A169813 a(n) = n XOR sigma(n), where sigma(n) is the number of divisors of n, A000203.

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A227320 Binary XOR of proper divisors of n.

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