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 1-4 of 4 results.

A227873 Sum of odious divisors of n. See A000069 for odious numbers.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 8, 15, 1, 3, 12, 7, 14, 24, 1, 31, 1, 3, 20, 7, 29, 36, 1, 15, 26, 42, 1, 56, 1, 3, 32, 63, 12, 3, 43, 7, 38, 60, 14, 15, 42, 87, 1, 84, 1, 3, 48, 31, 57, 78, 1, 98, 1, 3, 67, 120, 20, 3, 60, 7, 62, 96, 29, 127, 14, 36, 68, 7, 70, 129, 1, 15
Offset: 1

Views

Author

Vladimir Shevelev, Oct 25 2013

Keywords

Comments

Sum of evil divisors of n is A000203(n) - a(n) = A260934(n). See A001969 for evil numbers.

Links

Crossrefs

Cf. A000203, A227872, A000069, A001969, A212302, A260934, A010060, A102392.

Programs

  • Maple
    A227873 := proc(n)
    option remember ;
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if not isA001969(d) then
            a := a+d ;
        end if;
    end do:
    a ;
end proc:
seq(A227873(n),n=1..200) ; # R. J. Mathar, Aug 17 2022
  • Mathematica
    Total[Select[Divisors@ #, OddQ@ First@ DigitCount[#, 2] &]] & /@ Range@ 72 (* Michael De Vlieger, Aug 04 2015 *)
  • PARI
    a(n) = sumdiv(n, d, d*(hammingweight(d) % 2)); \\ Michel Marcus, Aug 04 2015

Formula

a(n) = Sum_{d|n} A102392(d). - Ridouane Oudra, Apr 19 2025

Extensions

More terms from Peter J. C. Moses
Minor changes. - Wolfdieter Lang, Aug 23 2015

A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70
Offset: 1

Views

Author

Amiram Eldar, Nov 21 2023

Keywords

Comments

First differs from A056192 at n = 32, and from A270418 and A367168 at n = 128.

Links

Crossrefs

Cf. A000069, A001221, A010060, A034444, A102392, A262675, A270428, A293439, A366901, A366903, A367515.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A366905, A367168, A367513.

Programs

  • Mathematica
    f[p_, e_] := p^(e*ThueMorse[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(hammingweight(f[i, 2])%2, f[i, 1]^f[i, 2], 1));}
  • Python
    from math import prod
from sympy import factorint
def A367514(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Formula

Multiplicative with a(p^e) = p^(e*A010060(e)) = p^A102392(e).
a(n) = n/A367513(n).
A001221(a(n)) = A293439(n).
A034444(a(n)) = A367515(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= n, with equality if and only if n is an exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.88585652437242918295..., and f(x) = (x+2)/(2*(x+1)) + (x/2) * Product_{k>=0} (1 - x^(2^k)).

A102391 Evil numbers in evil places.

Original entry on oeis.org

0, 0, 0, 3, 0, 5, 6, 0, 0, 9, 10, 0, 12, 0, 0, 15, 0, 17, 18, 0, 20, 0, 0, 23, 24, 0, 0, 27, 0, 29, 30, 0, 0, 33, 34, 0, 36, 0, 0, 39, 40, 0, 0, 43, 0, 45, 46, 0, 48, 0, 0, 51, 0, 53, 54, 0, 0, 57, 58, 0, 60, 0, 0, 63, 0, 65, 66, 0, 68, 0, 0, 71, 72, 0, 0, 75, 0, 77, 78, 0, 80, 0, 0, 83, 0, 85
Offset: 0

Views

Author

Paul Barry, Jan 06 2005

Keywords

Comments

Evil numbers A001969(n) appear at positions indexed by the evil numbers, 0 otherwise. A001477(n) = A102391(n) + A102392(n).

Crossrefs

Cf. A000120, A001477, A001969, A010060, A102391, A102392.

Programs

  • Mathematica
    a[n_] := If[EvenQ @ DigitCount[n, 2, 1], n, 0]; Array[a, 100, 0] (* Amiram Eldar, Aug 02 2020 *)
  • Python
    def A102391(n): return 0 if n.bit_count()&1 else n # Chai Wah Wu, Nov 23 2023

Formula

a(n) = if((1+floor(n/2)) (1+(-1)^A000120(n))/2 = 0, 0, n).
a(n) = if((1+floor(n/2)) (1+(-1)^A010060(n))/2 = 0, 0, n).

A102394 A wicked odious sequence.

Original entry on oeis.org

0, 2, 3, 0, 5, 0, 0, 8, 9, 0, 0, 12, 0, 14, 15, 0, 17, 0, 0, 20, 0, 22, 23, 0, 0, 26, 27, 0, 29, 0, 0, 32, 33, 0, 0, 36, 0, 38, 39, 0, 0, 42, 43, 0, 45, 0, 0, 48, 0, 50, 51, 0, 53, 0, 0, 56, 57, 0, 0, 60, 0, 62, 63, 0, 65, 0, 0, 68, 0, 70, 71, 0, 0, 74, 75, 0, 77, 0, 0, 80, 0, 82, 83, 0, 85, 0
Offset: 0

Views

Author

Paul Barry, Jan 06 2005

Keywords

Comments

Odious numbers plus one (A000069(n)+1) appear at positions indexed by the odious numbers, 0 otherwise. A000027(n) = A102393(n) + A102394(n).

Crossrefs

Cf. A000027, A000069, A000120, A010060, A102392, A102393.

Programs

  • Mathematica
    a[n_] := If[OddQ @ DigitCount[n, 2, 1], n + 1, 0]; Array[a, 100, 0] (* Amiram Eldar, Aug 02 2020 *)

Formula

a(n) = (n+1)(1-(-1)^A000120(n))/2.
a(n) = (n+1)(1-(-1)^A010060(n))/2.
Showing 1-4 of 4 results.

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