cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A093688 Numbers m such that all divisors of m, excluding the divisor 1, have an even number of 1's in their binary expansions.

Original entry on oeis.org

1, 3, 5, 9, 15, 17, 23, 27, 29, 43, 45, 51, 53, 71, 83, 85, 89, 101, 113, 129, 135, 139, 149, 153, 159, 163, 197, 215, 249, 255, 257, 263, 267, 269, 277, 281, 293, 303, 311, 317, 337, 347, 349, 353, 359, 373, 383, 387, 389, 401, 417, 447, 449, 459, 461, 467, 479
Offset: 1

Views

Author

Jason Earls, May 16 2004

Keywords

Comments

Putting the 1 aside, these could be called odiousfree numbers, and are a subsequence of A001969. A093696 would be the evilfree numbers then. - Irina Gerasimova, Feb 08 2014.
Equivalently, numbers all of whose divisors > 1 are evil (A001969). - Bernard Schott, Jul 24 2022

Examples

			51 is in the sequence because, excluding 1, its divisors are 3, 17 and 51.
In binary: 11, 10001, 110011 all have an even number of 1's.

Links

Crossrefs

Cf. A001969, A093696, A356018, A356019, A356020.

Programs

  • Mathematica
    okQ[n_] := AllTrue[Rest[Divisors[n]], EvenQ[Total[IntegerDigits[#, 2]]]&]; Select[Range[500], okQ] (* Jean-François Alcover, Dec 06 2015 *)
  • PARI
    is(n)=sumdiv(n,d,hammingweight(d)%2)==1 \\ Charles R Greathouse IV, Mar 28 2013
  • Python
    from sympy import divisors
def c(n): return n == 1 or bin(n).count("1")&1 == 0
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(480) if ok(k)]) # Michael S. Branicky, Jul 24 2022

Extensions

a(1) added by Charles R Greathouse IV, Mar 28 2013

A356018 a(n) is the number of evil divisors (A001969) of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 0, 2, 2, 0, 3, 0, 0, 3, 0, 1, 4, 0, 3, 1, 0, 1, 4, 1, 0, 3, 0, 1, 6, 0, 0, 2, 2, 1, 6, 0, 0, 2, 4, 0, 2, 1, 0, 5, 2, 0, 5, 0, 2, 3, 0, 1, 6, 1, 0, 2, 2, 0, 9, 0, 0, 3, 0, 2, 4, 0, 3, 2, 2, 1, 8, 0, 0, 4, 0, 1, 4, 0, 5, 3, 0, 1, 3, 3, 2
Offset: 1

Views

Author

Bernard Schott, Jul 23 2022

Keywords

Comments

a(n) = 0 iff n is in A093696.

Examples

			12 has 6 divisors: {1, 2, 3, 4, 6, 12} of which three {3, 6, 12} have an even number of 1's in their binary expansion with 11, 110 and 11100 respectively; hence a(12) = 3.

Crossrefs

Cf. A000005, A001969, A093688, A093696 (location of 0s), A227872, A356019, A356020.
Similar sequences: A083230, A087990, A087991, A332268, A355302.

Programs

  • Maple
    A356018 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isA001969(d) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A356018(n),n=1..200) ;  # R. J. Mathar, Aug 07 2022
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)
  • PARI
    a(n) = my(v = valuation(n, 2)); n>>=v; d=divisors(n); sum(i=1, #d, bitand(hammingweight(d[i]), 1) == 0) * (v+1) \\ David A. Corneth, Jul 23 2022
  • Python
    from sympy import divisors
def c(n): return bin(n).count("1")&1 == 0
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 23 2022

Formula

a(n) = A000005(n) - A227872(n).

Extensions

More terms from David A. Corneth, Jul 23 2022

A356020 Positions of records in A356018, i.e., integers whose number of evil divisors sets a new record.

Original entry on oeis.org

1, 3, 6, 12, 18, 30, 60, 90, 120, 180, 360, 540, 720, 1080, 1440, 2160, 3780, 4320, 6120, 7560, 8640, 12240, 15120, 24480, 27720, 30240, 36720, 48960, 50400, 55440, 73440, 83160, 110880, 128520, 138600, 166320, 221760, 257040, 277200, 332640, 471240, 514080, 554400
Offset: 1

Views

Author

Bernard Schott, Jul 24 2022

Keywords

Comments

Corresponding records of number of evil divisors are 0, 1, 2, 3, 4, 6, 9, 10, 12, 15, ...

Examples

			60 is in the sequence because A356018(60) = 9 is larger than any earlier value in A356018.

Links

Crossrefs

Cf. A001969, A093688, A093696, A356018, A356019.
Similar sequences: A093036, A093037, A330815, A350756, A355969.

Programs

  • Mathematica
    f[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 24 2022 *)
  • PARI
    upto(n) = my(res = List(), r=-1); forfactored(i=1, n, if(numdiv(i[2]) > r, d = divisors(i[2]); c=sum(j=1, #d, isevil(d[j])); if(c>r, r=c; listput(res,i[1])))); res
isevil(n) = bitand(hammingweight(n), 1)==0 \\ David A. Corneth, Jul 24 2022
  • Python
    from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1 == 0
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen(record=-1):
    for k in count(1):
        if f(k) > record: record = f(k); yield k
print(list(islice(agen(), 40))) # Michael S. Branicky, Jul 24 2022

Extensions

More terms from Amiram Eldar, Jul 24 2022
Showing 1-3 of 3 results.

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