A178910 -id:A178910 - cp's OEIS frontend

A235490 Numbers such that none of their prime factors share common 1-bits in the same bit-position and when added (or "ored" or "xored") together, yield a term of A000225 (a binary "repunit").

1, 3, 7, 10, 26, 31, 58, 122, 127, 1018, 2042, 8186, 8191, 32762, 131071, 524287, 2097146, 8388602, 33554426, 1073741818, 2147483647, 2305843009213693951, 618970019642690137449562111, 39614081257132168796771975162, 162259276829213363391578010288127, 166153499473114484112975882535043066

Offset: 1

Antti Karttunen, Jan 22 2014

a(1) = 1 is included on the grounds that it has no prime factors, thus A001414(1)=0, and 0 is one of the terms of A000225, marking the "repunit of length zero".

After 1, the sequence is a union of A000668 (Mersenne primes) and semiprimes of the form 2*A050415. The terms were constructed from the data given in those two entries.

			7 is included, because it is a prime, and repunit in base-2: '111'.
10 is included, as 10=2*5, and when we add 2 ('10' in binary) and 5 ('101' in binary), we also get 7 ('111' in binary), without producing any carries.

Subsequence of A235050.
Cf. A000225, A000668, A050415, A001414, A072593, A178910, A227843.
Cf. also A235488, A230492, A235032, A050376.

A323394 Carryless sum of divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 2, 8, 5, 3, 18, 12, 18, 14, 14, 14, 11, 18, 19, 10, 32, 22, 36, 24, 30, 21, 32, 20, 36, 20, 52, 32, 43, 48, 44, 38, 51, 38, 40, 46, 70, 42, 76, 44, 74, 58, 62, 48, 84, 47, 83, 62, 88, 54, 80, 62, 80, 60, 70, 50, 48, 62, 96, 84, 7, 74, 24, 68, 6

Offset: 1

Rémy Sigrist, Jan 13 2019

This sequence is a variant of A178910 for the base 10.

			For n = 42:
- the divisors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42,
- the sum of the units is: 1 + 2 + 3 + 6 + 7 + 4 + 1 + 2 = 26 == 6 (mod 10),
- the sum of the tens is: 1 + 2 + 4 = 7,
- hence a(42) = 76.
For n = 973:
- the divisors of 973 are: 1, 7, 139, 973,
- the sum of the units is: 1 + 7 + 9 + 3 = 20 == 0 (mod 10),
- the sum of the tens is: 3 + 7 = 10 == 0 (mod 10),
- the sum of the hundreds is: 1 + 9 = 10 == 0 (mod 10),
- hence a(973) = 0.

Rémy Sigrist, Table of n, a(n) for n = 1..19999
Rémy Sigrist, Scatterplot of the first 1000000 terms
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to sigma(n)

Cf. A000203, A169890, A178910, A323414 (positions of zeros), A323415 (fixed points).

f:= proc(n) local t,d,dd,m,i;
t:= Vector(convert(n,base,10));
for d in numtheory:-divisors(n) minus {n} do
  dd:= convert(d,base,10);
  m:= nops(dd);
  t[1..m]:= t[1..m] + Vector(dd) mod 10;
od:
add(t[i]*10^(i-1),i=1..ilog10(n)+1)
end proc:
map(f, [$1..100]); # Robert Israel, Jan 15 2019

a(n, base=10) = my (v=[]); fordiv (n, d, my (w=Vecrev(digits(d, base))); v=vector(max(#v, #w), k, (if (k>#v, w[k], k>#w, v[k], (v[k]+w[k])%base)))); fromdigits(Vecrev(v), base)

a(n) <= A000203(n).

A328178 a(n) is the minimal value of the expression d XOR (n/d) where d runs through the divisors of n and XOR denotes the bitwise XOR operator.

Original entry on oeis.org

0, 3, 2, 0, 4, 1, 6, 6, 0, 7, 10, 4, 12, 5, 6, 0, 16, 5, 18, 1, 4, 9, 22, 2, 0, 15, 10, 3, 28, 3, 30, 12, 8, 19, 2, 0, 36, 17, 14, 13, 40, 1, 42, 15, 12, 21, 46, 8, 0, 15, 18, 9, 52, 15, 14, 10, 16, 31, 58, 9, 60, 29, 14, 0, 8, 13, 66, 21, 20, 11, 70, 1, 72

Offset: 1

Rémy Sigrist, Oct 06 2019

			For n = 12:
- we have the following values:
    d   12/d  d XOR (12/d)
    --  ----  ------------
     1    12            13
     2     6             4
     3     4             7
     4     3             7
     6     2             4
    12     1            13
- hence a(12) = min({4, 7, 13}) = 4.

Rémy Sigrist, Table of n, a(n) for n = 1..16384
Rémy Sigrist, Logarithmic scatterplot of (n, 1+a(n)) for n = 1..2^16

See A328176 and A328177 for similar sequences.

Cf. A178910.

a:= n-> min(seq(Bits[Xor](d, n/d), d=numtheory[divisors](n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2019

mvx[n_]:=Min[BitXor[#,n/#]&/@Divisors[n]]; Array[mvx,80] (* Harvey P. Dale, Nov 04 2019 *)

a(n) = vecmin(apply(d -> bitxor(d, n/d), divisors(n)))

a(n) = 0 iff n is a square.

a(p) = p-1 for any odd prime number p.

A178911 Perfex numbers: n = binary XOR of divisors of n.

Original entry on oeis.org

1, 6, 120, 198, 3696, 6240, 32640, 56160, 1941408, 3592200, 8119800, 15628032, 27125280, 59032080, 61788240, 125859840, 1635834720, 2147450880, 3709081680, 16328199552, 26198072160, 52344970080, 52396088160, 209584184160, 210197601120, 236223190200, 237385437360

Offset: 1

Franklin T. Adams-Watters, Jun 22 2010

a(17) > 1e9.
10^11 < a(24) <= 209584184160. a(25) <= 210197601120. - Donovan Johnson, Mar 12 2011
a(28) > 3*10^11. - Giovanni Resta, Aug 14 2019

Cf. A178910, A178909, A000396, A003987.

lst = {}; k = 1; While[k < 10^9, If[ BitXor @@ Divisors@k == k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Jun 27 2010 *)

xigma(n)=local(ds,r);ds=divisors(n);for(k=1,#ds,r=bitxor(r,ds[k]));r
for(n=1,1000000000,if(xigma(n)==n,print1(n",")))

a(17) from Robert G. Wilson v, Jul 30 2010
a(18)-a(23) from Donovan Johnson, Mar 12 2011
a(24)-a(27) from Giovanni Resta, Aug 14 2019

A293212 Binary XOR of prime divisors of n.

Original entry on oeis.org

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

Offset: 2

Alex Ratushnyak, Feb 04 2018

The sequence of indices of zeros begins: 70, 140, 280, 350, 490, 560, 646, 700, 980, 1120, 1292, 1400, 1750, 1798, 1960, 2145.

			a(6) = a(24) = 2 XOR 3 = 1.
a(2145) = 3 XOR 5 XOR 11 XOR 13 = 0.

Alois P. Heinz, Table of n, a(n) for n = 2..20000

Cf. A000040, A072594, A178910.

a:= proc(n) local d, r; r:=0; for d in numtheory
      [factorset](n) do r:= Bits[Xor](r, d) od; r
    end:
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 09 2018

a(n) = my(vp = factor(n)[,1]~, k=0); for (i=1, #vp, k = bitxor(k, vp[i])); k; \\ Michel Marcus, Feb 05 2018

from functools import reduce
from operator import xor
from sympy import primefactors
def A293212(n): return reduce(xor,primefactors(n)) # Chai Wah Wu, Jun 03 2025

a(n) = n iff n is a prime.

A318504 SumXOR of divisors of n, up to, but not including the second largest of them A032742(n); a(1) = 0 by convention.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 3, 1, 3, 0, 4, 0, 3, 2, 7, 0, 6, 0, 2, 2, 3, 0, 10, 1, 3, 2, 0, 0, 9, 0, 15, 2, 3, 4, 7, 0, 3, 2, 0, 0, 15, 0, 12, 14, 3, 0, 22, 1, 12, 2, 10, 0, 29, 4, 6, 2, 3, 0, 26, 0, 3, 12, 31, 4, 27, 0, 22, 2, 5, 0, 5, 0, 3, 8, 20, 6, 17, 0, 4, 11, 3, 0, 14, 4, 3, 2, 18, 0, 3, 6, 16, 2, 3, 4, 46, 0, 10, 0, 5, 0, 53, 0, 24, 26

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A003987, A032742, A106409, A178910, A227320, A318505, A318507, A318517.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318504(n) = { my(v=0); fordiv(n,d,if(d<A032742(n), v = bitxor(v,d))); (v); };

a(n) = A032742(n) XOR A227320(n).

For n > 1, a(n) = A106409(n) XOR A178910(n).

A227443 Numbers k such that k and k+1 have the same binary XOR of divisors.

Original entry on oeis.org

6, 110, 116, 194, 198, 204, 237, 437, 452, 507, 965, 969, 986, 1011, 1736, 1748, 1808, 1824, 1896, 1971, 1989, 2021, 3696, 6230, 6243, 6533, 6875, 12974, 13736, 13764, 14567, 14790, 15720, 26625, 29516, 29715, 32102, 49251, 55686, 57489, 61329, 64440, 99780, 104247, 114738

Offset: 1

Alex Ratushnyak, Jul 11 2013

n such that A178910(n+1) = A178910(n).

Amiram Eldar, Table of n, a(n) for n = 1..700

Cf. A178910, A002961.

b[n_] := BitXor @@ Divisors[n]; b1 = 0; s = {}; Do[b2 = b[n]; If[b1 == b2, AppendTo[s, n-1]]; b1 = b2, {n, 1, 10^4}]; s (* Amiram Eldar, Sep 19 2019 *)

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

A384441 Binary XOR of n and the prime factors of n.

Original entry on oeis.org

1, 0, 0, 6, 0, 7, 0, 10, 10, 13, 0, 13, 0, 11, 9, 18, 0, 19, 0, 19, 17, 31, 0, 25, 28, 21, 24, 25, 0, 26, 0, 34, 41, 49, 33, 37, 0, 55, 41, 47, 0, 44, 0, 37, 43, 59, 0, 49, 54, 53, 33, 59, 0, 55, 57, 61, 41, 37, 0, 56, 0, 35, 59, 66, 73, 72, 0, 87, 81, 70, 0, 73, 0, 109

Offset: 1

Karl-Heinz Hofmann, May 30 2025

			For n = 12 the prime factors are {2,3} -> a(12) = 12 XOR 2 XOR 3 = 13.
a(13) = 13 XOR 13 = 0.

Alois P. Heinz, Table of n, a(n) for n = 1..16384
Karl-Heinz Hofmann, Graph up to n = 2^17
Karl-Heinz Hofmann, Zoom trip of the graph with number of prime factors of n colored.
Wikipedia, Bitwise operation: XOR

Cf. A000040, A052548, A178910, A293212.

f:= l-> `if`(l=[], 0, Bits[Xor](l[1], f(l[2..-1]))):
a:= n-> f([n, map(i-> i[1], ifactors(n)[2])[]]):
seq(a(n), n=1..74);  # Alois P. Heinz, May 30 2025

a[n_] := BitXor @@ Join[{n}, FactorInteger[n][[;; , 1]]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, May 30 2025 *)

a(n) = my(f=factor(n)[,1]); my(b=n); for (k=1, #f, b=bitxor(b, f[k])); b; \\ Michel Marcus, May 30 2025

from sympy import primefactors
def A384441(n):
    result = n
    for pf in primefactors(n): result ^= pf
    return result

a(n) = XOR(n,A293212(n)).
a(n) = 0 <=> n is prime.
a(2^n) = A052548(n) for n>=2.

cp's OEIS Frontend

A235490 Numbers such that none of their prime factors share common 1-bits in the same bit-position and when added (or "ored" or "xored") together, yield a term of A000225 (a binary "repunit").

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A323394 Carryless sum of divisors of n.

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A328178 a(n) is the minimal value of the expression d XOR (n/d) where d runs through the divisors of n and XOR denotes the bitwise XOR operator.

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A178911 Perfex numbers: n = binary XOR of divisors of n.

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A293212 Binary XOR of prime divisors of n.

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A318504 SumXOR of divisors of n, up to, but not including the second largest of them A032742(n); a(1) = 0 by convention.

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A227443 Numbers k such that k and k+1 have the same binary XOR of divisors.

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Mathematica

A331700 Binary XOR of squares of divisors of n.

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