A072594 -id:A072594 - cp's OEIS frontend

A072595 Numbers k such that A072594(k) = 0.

4, 9, 16, 25, 36, 49, 64, 70, 81, 100, 121, 144, 169, 196, 225, 256, 280, 289, 324, 361, 400, 441, 484, 529, 576, 625, 630, 646, 676, 729, 784, 841, 900, 961, 1024, 1089, 1120, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1750, 1764, 1798, 1849, 1936

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

All squares are terms, but also 70 is a term, see example, A072596.

			A072594(70) = A072594(2*5*7) = A072594(2) 'xor' A072594(5) 'xor' A072594(7) = '010' xor '101' xor '111' = '000' = 0.

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

Cf. A000290.

a072595 n = a072595_list !! (n-1)
a072595_list = filter ((== 0) . a072594) [1..]
-- Reinhard Zumkeller, Nov 17 2012

A072596 Nonsquares with A072594(n) = 0.

Original entry on oeis.org

70, 280, 630, 646, 1120, 1750, 1798, 2145, 2520, 2584, 3430, 3526, 4480, 5670, 5814, 5865, 6006, 7000, 7192, 8470, 8580, 9177, 10080, 10336, 11305, 11830, 13110, 13720, 14104, 15750, 16150, 16182, 16422, 17920, 19305, 20230, 20553, 20806

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

A010052(a(n)) = 0. - Reinhard Zumkeller, Nov 17 2012

Reinhard Zumkeller and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zumkeller)

Cf. A072595.

a072596 n = a072596_list !! (n-1)
a072596_list = filter ((== 0) . a010052) a072595_list
-- Reinhard Zumkeller, Nov 17 2012

is(n)=if(issquare(n), return(0)); my(f=factor(n), t); for(i=1, #f~, if(f[i, 2]%2, t=bitxor(t, f[i, 1]))); t==0 \\ Charles R Greathouse IV, Aug 28 2016

A127812 Smallest m such that A072594(m) = n.

Original entry on oeis.org

4, 1, 2, 3, 21, 5, 15, 7, 33, 22, 66, 11, 77, 13, 39, 26, 57, 17, 51, 19, 69, 46, 95, 23, 145, 465, 155, 435, 93, 29, 87, 31, 185, 777, 259, 555, 222, 37, 111, 74, 129, 41, 123, 43, 141, 94, 215, 47, 265, 1113, 371, 795, 318, 53, 159, 106, 177, 118, 354, 59, 366, 61

Offset: 0

Reinhard Zumkeller, Feb 02 2007

nonn

A072594(a(n)) = n and A072594(m) <> n for m < a(n).

R. Zumkeller, Table of n, a(n) for n = 0..1000

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a127812 = (+ 1) . fromJust . (`elemIndex` a072594_list)
-- Reinhard Zumkeller, Nov 17 2012

A178910 Binary XOR of divisors of n.

Original entry on oeis.org

1, 3, 2, 7, 4, 6, 6, 15, 11, 12, 10, 14, 12, 10, 8, 31, 16, 29, 18, 28, 16, 30, 22, 30, 29, 20, 16, 18, 28, 24, 30, 63, 40, 48, 32, 49, 36, 54, 40, 60, 40, 48, 42, 54, 44, 58, 46, 62, 55, 39, 32, 36, 52, 48, 56, 34, 40, 36, 58, 56, 60, 34, 38, 127, 72, 120, 66, 112, 80, 96, 70

Offset: 1

Franklin T. Adams-Watters, Jun 22 2010

If 2^k <= n < 2^(k+1), then also 2^k <= a(n) < 2^(k+1), since any proper divisor of n is < 2^k.

Reinhard Zumkeller, Table of n, a(n) for n = 1..8191

Cf. A000203, A178908, A178911, A003987.

Cf. A027750, A072594; subsequences A028982 (odd), A028982 (even).

import Data.Bits (xor)
a178910 = foldl1 xor . a027750_row :: Integer -> Integer
-- Reinhard Zumkeller, Nov 17 2012

a(n)=local(ds,r);ds=divisors(n);for(k=1,#ds,r=bitxor(r,ds[k]));r

from sympy import divisors
def A178910(n):
    res = 1
    for divisor in divisors(n)[1:]: res ^= divisor
    return res # Karl-Heinz Hofmann, May 30 2025

A072593 In prime factorization of n replace multiplication with bitwise logical 'or'.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 7, 11, 3, 13, 7, 7, 2, 17, 3, 19, 7, 7, 11, 23, 3, 5, 15, 3, 7, 29, 7, 31, 2, 11, 19, 7, 3, 37, 19, 15, 7, 41, 7, 43, 11, 7, 23, 47, 3, 7, 7, 19, 15, 53, 3, 15, 7, 19, 31, 59, 7, 61, 31, 7, 2, 13, 11, 67, 19, 23, 7, 71, 3, 73, 39, 7, 19, 15, 15, 79, 7, 3, 43, 83

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

a(n) = a(A007947(n)); n is prime iff a(n)=n.

			a(35) = a(5*7) = a(5) 'or' a(7) = '101' or '111' = '111' = 7.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A072591, A072594.

Cf. A027746.

import Data.Bits (((.|.))
a072593 = foldl1 (.|.) . a027746_row  -- Reinhard Zumkeller, Jul 05 2013

Array[BitOr @@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, 120] (* Michael De Vlieger, May 31 2025 *)

a072593(n) = if(n<2, return(1)); my(F=factor(n), v=Vec(F[,1]), x=v[1]); for(k=2, #v, x=bitor(x,v[k])); x \\ Hugo Pfoertner, May 31 2025

from sympy import factorint
from operator import _or_
from functools import reduce
def a(n): return reduce(_or_, (f for f in factorint(n))) if n > 1 else 1
print([a(n) for n in range(1, 84)]) # Michael S. Branicky, May 31 2025

A072591 In prime factorization of n replace multiplication with bitwise logical 'and'.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 0, 11, 2, 13, 2, 1, 2, 17, 2, 19, 0, 3, 2, 23, 2, 5, 0, 3, 2, 29, 0, 31, 2, 3, 0, 5, 2, 37, 2, 1, 0, 41, 2, 43, 2, 1, 2, 47, 2, 7, 0, 1, 0, 53, 2, 1, 2, 3, 0, 59, 0, 61, 2, 3, 2, 5, 2, 67, 0, 3, 0, 71, 2, 73, 0, 1, 2, 3, 0, 79, 0, 3, 0, 83, 2, 1, 2, 1, 2, 89, 0, 5, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

a(n) = a(A007947(n)); n is prime iff a(n)=n;

a(n)=0 iff n is even and one prime factor is of form 4*k+1.

			a(35) = a(5*7) = a(5) 'and' a(7) = '101' and '111' = '101' = 5.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A072593, A072594, A072592.

Cf. A027746.

import Data.Bits ((.&.))
a072591 = foldl1 (.&.) . a027746_row  -- Reinhard Zumkeller, Jul 05 2013

a[n_] := BitAnd @@ FactorInteger[n][[All, 1]];
Array[a, 100] (* Jean-François Alcover, Nov 16 2021 *)

from sympy import factorint
from operator import _and_
from functools import reduce
def a(n): return reduce(_and_, (f for f in factorint(n))) if n > 1 else 1
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, May 31 2025

A235050 Squarefree numbers such that none of their prime factors share common 1-bits in the same bit-position of their binary representations.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 17, 19, 23, 26, 29, 31, 34, 37, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 74, 79, 82, 83, 89, 97, 101, 103, 106, 107, 109, 113, 122, 127, 131, 137, 139, 146, 149, 151, 157, 163, 167, 173, 178, 179, 181, 191, 193, 194, 197, 199, 202, 211, 218, 223, 226, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Antti Karttunen, Jan 22 2014

a(1)=1 is included on the grounds that it has no prime factors at all, thus no conflicting 1-bits.
After a(1)=1 all n such that A001414(n) = A072593(n), or equally, A001414(n) = A072594(n).
Union of noncomposites (A008578) and semiprimes of the form 2*A002144 (cf. also A235490).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequences: A000040, A235490.
Subsequence of A005117.
Cf. also A001414, A072593, A072594, A235040.

A235488 Squarefree numbers which yield zero when their prime factors are xored together.

Original entry on oeis.org

70, 646, 1798, 2145, 3526, 5865, 6006, 9177, 11305, 13110, 16422, 20553, 20806, 21489, 23529, 28905, 28985, 30305, 31465, 37961, 38086, 38454, 42441, 44022, 44998, 45353, 45942, 46345, 53985, 54230, 55913, 60630, 60697, 61705, 62049, 64790, 78406, 80934, 81158

Offset: 1

Antti Karttunen, Jan 22 2014

All n for which A008683(n) <> 0 and A072594(n) = 0.
It seems that an analogous case as A072595 for GF(2)[X]-polynomials is just the squares of GF(2)[X]-polynomials (A000695), thus in that ring, the sequence analogous to this one would be empty.
This sequence happens also to encode in the prime factorization of n a certain subset of the Nim game positions that are second-player win.

			70 is included, as 70 = 2*5*7, whose binary representations are '10', '101' and '111', which when all are xored (cf. A003987) together, cancel all 1-bits, thus yielding zero.
212585 is included, as 212585 = 5*17*41*61, and when we xor their base-2 representations together:
     101
   10001
  101001
  111101
--------
  000000
we get only zeros, because in each column (bit-position), there is an even number of 1-bits.

Antti Karttunen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 75 terms from Karttunen)
Wikipedia, Nim

Intersection of A005117 and A072595 (equally: of A005117 and A072596).

Cf. A003987, A235490, A079599, A048833, A050314.

Select[Range[82000],SquareFreeQ[#]&&BitXor@@FactorInteger[#][[All,1]]==0&] (* Harvey P. Dale, Apr 01 2017 *)

is(n)=if(n<9, return(0)); my(f=factor(n)); vecmax(f[,2])==1 && fold(bitxor, f[,1])==0 \\ Charles R Greathouse IV, Aug 06 2016

A260737 Sum of Hamming distances between binary representations of prime factors of n, summed over all nonordered pairs of primes present (with multiplicity) in the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 2, 2, 0, 0, 2, 0, 6, 1, 2, 0, 3, 0, 4, 0, 4, 0, 6, 0, 0, 1, 3, 1, 4, 0, 2, 3, 9, 0, 4, 0, 4, 4, 3, 0, 4, 0, 6, 2, 8, 0, 3, 3, 6, 1, 5, 0, 10, 0, 4, 2, 0, 1, 4, 0, 6, 2, 6, 0, 6, 0, 4, 4, 4, 2, 8, 0, 12, 0, 4, 0, 7, 2, 3, 4, 6, 0, 9, 2, 6, 3, 4, 3, 5, 0, 4, 2, 12, 0, 6, 0, 12, 4, 5, 0, 6, 0, 8, 3, 8, 0, 4, 2, 10, 6, 4, 3, 14

Offset: 1

Antti Karttunen, Sep 22 2015

			For n = 1 the prime factorization is empty, thus there is nothing to sum, so a(1) = 0.
For n = 6 = 2*3, a(6) = 1 because the Hamming distance between 2 and 3 is 1 as 2 = "10" in binary and 3 = "11" in binary.
For n = 10 = 2*5, a(10) = 3 because the Hamming distance between 2 and 5 is 3 as 2 = "10" in binary (extended with a leading zero to make it "010") and 5 = "101" in binary.
For n = 12 = 2*2*3, a(12) = 2 because the Hamming distance between 2 and 3 is 1, and the pair (2,3) occurs twice as one can pick either one of the two 2's present in the prime factorization to be a pair of a single 3. Note that the Hamming distance between 2 and 2 is 0, thus the pair (2,2) of prime divisors does not contribute to the sum.
For n = 36 = 2*2*3*3, a(36) = 4 because the Hamming distance between 2 and 3 is 1, and the prime factor pair (2,3) occurs four times in total. Note that the Hamming distance is zero between 2 and 2 as well as between 3 and 3, thus the pairs (2,2) and (3,3) do not contribute to the sum.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A101080.
Cf. A000961 (positions of the zeros), A261077 (positions of the ones).
Cf. A072594, A072595, A235488.
Cf. also A261079.

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.

cp's OEIS Frontend

A072595 Numbers k such that A072594(k) = 0.

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A072596 Nonsquares with A072594(n) = 0.

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A127812 Smallest m such that A072594(m) = n.

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A178910 Binary XOR of divisors of n.

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A072593 In prime factorization of n replace multiplication with bitwise logical 'or'.

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A072591 In prime factorization of n replace multiplication with bitwise logical 'and'.

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Haskell

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A235050 Squarefree numbers such that none of their prime factors share common 1-bits in the same bit-position of their binary representations.

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A235488 Squarefree numbers which yield zero when their prime factors are xored together.

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