A072593 -id:A072593 - 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

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.

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").

Original entry on oeis.org

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.

cp's OEIS Frontend

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

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

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Haskell

Mathematica

Python

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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Author

Keywords

Comments

Links

Crossrefs

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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