A072591 -id:A072591 - 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

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

A072592 Even numbers with at least one prime factor of form 4*k+1.

Original entry on oeis.org

10, 20, 26, 30, 34, 40, 50, 52, 58, 60, 68, 70, 74, 78, 80, 82, 90, 100, 102, 104, 106, 110, 116, 120, 122, 130, 136, 140, 146, 148, 150, 156, 160, 164, 170, 174, 178, 180, 182, 190, 194, 200, 202, 204, 208, 210, 212, 218, 220, 222, 226, 230, 232, 234, 238

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

Conjecture: this is exactly the sequence whose terms are twice those of A009003. (This has been verified for all terms<=500.) Compare A009003. - John W. Layman, Mar 12 2008

The conjecture is true. See comments on A008846 and A004613. - Lambert Herrgesell (zero815(AT)googlemail.com), Apr 24 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005843, A002144, A072591, A009003, A008846, A004613.

opfQ[n_]:=Count[Transpose[FactorInteger[n]][[1]],?(IntegerQ[ (#-1)/4]&)]>0; Select[Range[2,250,2],opfQ] (* _Harvey P. Dale, Jun 02 2012 *)

from itertools import count, islice
from sympy import primefactors
def A072592_gen(startvalue=1): # generator of terms >= startvalue
    return (n<<1 for n in count(max(startvalue,1)) if any(p&3==1 for p in primefactors(n)))
A072592_list = list(islice(A072592_gen(),20)) # Chai Wah Wu, Jul 07 2024

A072591(a(n)) = 0.

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

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

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Author

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Comments

Examples

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Programs

Haskell

Mathematica

PARI

Python

A072592 Even numbers with at least one prime factor of form 4*k+1.

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