A072595 -id:A072595 - cp's OEIS frontend

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

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

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.

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

cp's OEIS Frontend

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

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

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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.

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A072596 Nonsquares with A072594(n) = 0.

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