A162644 -id:A162644 - cp's OEIS frontend

A340818 Numerators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

5, 7, 41, 3, 197, 229, 5827, 277, 1157, 8382, 268049, 94175911, 964941119, 1929224113, 31529606831, 835346466959, 3398377571053, 52665885581009, 119955940157647877, 34063199364211668943, 315047077264055066629, 199089493729235251718903, 47411489829747180146759

Offset: 1

Amiram Eldar, Jan 22 2021

Let Omega_n(k) be the number of prime divisors of k not exceeding prime(n) counted with multiplicity, and omega_n(k) the number of distinct prime divisors of k not exceeding prime(n). Then, f(n) = a(n)/A340819(n) is the asymptotic density of numbers k such that Omega_n(k) == omega_n(k) (mod 2).

Equivalently, f(n) is the asymptotic density of numbers k such that A046660(d_n(k)) is even, where d_n(k) is the largest prime(n)-smooth divisor of k.

			The sequence of fractions begins with 5/6, 7/9, 41/54, 3/4, 197/264, 229/308, 5827/7854, 277/374, 1157/1564, 8382/11339, ...
For n=1, Omega_2(k)-omega_2(k) is even for either odd k (A005408), or even k whose binary representation ends in an odd number of zeros (A036554). The disjoint union of these 2 sequences has an asymptotic density 1/2 + 1/3 = 5/6.

Amiram Eldar, Table of n, a(n) for n = 1..462
Jérémie Detrey, Pierre-Jean Spaenlehauer and Paul Zimmermann, Computing the rho constant, 2016.
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Cf. A005408, A036554, A046660, A162644, A340819 (denominators), A340820.

d[p_] := 1/(p*(p + 1)); delta[n_] := delta[n] = d[Prime[n]]; f[0] = 1; f[n_] := f[n] = f[n - 1] * (1 - delta[n]) + (1 - f[n - 1]) * delta[n]; Numerator @ Array[f, 30]

Let delta(n) = 1/(prime(n)*(prime(n)+1)) be the asymptotic density of numbers whose prime(n)-adic valuation is positive and even. Let f(0) = 1. Then, f(n) = f(n-1)*(1 - delta(n)) + (1 - f(n))*delta(n).

Limit_{n->oo} f(n) = 0.73584... (A340820).

A340819 Denominators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

Original entry on oeis.org

6, 9, 54, 4, 264, 308, 7854, 374, 1564, 11339, 362848, 127541072, 1307295988, 2614591976, 42742894912, 1132686715168, 4608863185856, 71437379380768, 162734350229389504, 46216555465146619136, 427503138052606227008, 270181983249247135469056, 64347502466822129824768

Offset: 1

Amiram Eldar, Jan 22 2021

See A340818 for details.

Amiram Eldar, Table of n, a(n) for n = 1..462

Cf. A046660, A162644, A340818 (numerators), A340820.

d[p_] := 1/(p*(p + 1)); delta[n_] := delta[n] = d[Prime[n]]; f[0] = 1; f[n_] := f[n] = f[n - 1] * (1 - delta[n]) + (1 - f[n - 1]) * delta[n]; Denominator @ Array[f, 30]

A036537 Numbers whose number of divisors is a power of 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Labos Elemer

nonn

Primes and A030513(d(x)=4) are subsets; d(16k+4) and d(16k+12) have the form 3Q, so x=16k+4 or 16k-4 numbers are missing.
A number m is a term if and only if all its divisors are infinitary, or A000005(m) = A037445(m). - Vladimir Shevelev, Feb 23 2017
All exponents in the prime number factorization of a(n) have the form 2^k-1, k >= 1. So it is an S-exponential sequence (see Shevelev link) with S={2^k-1}. Using Theorem 1, we obtain that a(n) ~ C*n, where C = Product((1-1/p)*(1 + Sum_{i>=1} 1/p^(2^i-1))). - Vladimir Shevelev Feb 27 2017
This constant is C = 0.687827... . - Peter J. C. Moses, Feb 27 2017
From Peter Munn, Jun 18 2022: (Start)
1 and numbers j*m^2, j squarefree, m >= 1, such that all prime divisors of m divide j, and m is in the sequence.
Equivalently, the nonempty set of numbers whose squarefree part (A007913) and squarefree kernel (A007947) are equal, and whose square part's square root (A000188) is in the set.
(End)

			383, 384, 385, 386 have 2, 16, 8, 4 divisors, respectively, so they are consecutive terms of this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithm., 175(2016), 385-395.
Eric Weisstein's World of Mathematics, Square Part, Squarefree Part

A005117, A030513, A058891, A175496, A336591 are subsequences.
Complement of A162643; subsequence of A002035. - Reinhard Zumkeller, Jul 08 2009
Subsequence of A162644, A337533.
Cf. A000005, A000188, A007913, A007947, A036538, A352780.
The closure of the squarefree numbers under application of A355038(.) and lcm.

a036537 n = a036537_list !! (n-1)
a036537_list = filter ((== 1) . a209229 . a000005) [1..]
-- Reinhard Zumkeller, Nov 15 2012

bi[ x_ ] := 1-Sign[ N[ Log[ 2, x ], 5 ]-Floor[ N[ Log[ 2, x ], 5 ] ] ]; ld[ x_ ] := Length[ Divisors[ x ] ]; Flatten[ Position[ Table[ bi[ ld[ x ] ], {x, 1, m} ], 1 ] ]
Select[Range[110],IntegerQ[Log[2,DivisorSigma[0,#]]]&] (* Harvey P. Dale, Nov 20 2016 *)

is(n)=n=numdiv(n);n>>valuation(n,2)==1 \\ Charles R Greathouse IV, Mar 27 2013

isok(m) = issquarefree(m) || (omega(m) == omega(core(m)) && isok(core(m,1)[2])); \\ Peter Munn, Jun 18 2022

from itertools import count, islice
from sympy import factorint
def A036537_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:not((k:=m+1)&-k)^k,factorint(n).values())),count(max(startvalue,1)))
A036537_list = list(islice(A036537_gen(),30)) # Chai Wah Wu, Jan 04 2023

A209229(A000005(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2012
a(n) << n. - Charles R Greathouse IV, Feb 25 2017
m is in the sequence iff for k >= 0, A352780(m, k+1) | A352780(m, k)^2. - Peter Munn, Jun 18 2022

A002035 Numbers that contain primes to odd powers only.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 101

Offset: 1

N. J. A. Sloane

Complement of the union of {1} and A072587. - Reinhard Zumkeller, Nov 15 2012, corrected version from Jun 23 2002
A036537 is a subsequence and this sequence is a subsequence of A162644. - Reinhard Zumkeller, Jul 08 2009
The asymptotic density of this sequence is A065463. - Amiram Eldar, Sep 18 2022

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eckford Cohen, Quadratic congruences with an odd number of summands, Amer. Math. Monthly, 73 (1966), 138-143.

Cf. A000203, A036537, A065463, A072586, A072587, A124010, A162644, A188999, A295316.

a002035 n = a002035_list !! (n-1)
a002035_list = filter (all odd . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 14 2012

isA002035 := proc(n)
    local pe;
    for pe in ifactors(n)[2] do
        if type(pe[2],'even') then
            return false;
        end if;
    end do:
    true ;
end proc:
A002035 := proc(n)
    option remember;
    if n =1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA002035(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002035(n),n=1..100) ; # R. J. Mathar, Nov 27 2017

ok[n_] := And @@ OddQ /@ FactorInteger[n][[All, 2]];
Select[Range[2, 101], ok]
(* Jean-François Alcover, Apr 22 2011 *)
Select[Range[2,110],AllTrue[FactorInteger[#][[All,2]],OddQ]&] (* Harvey P. Dale, Nov 02 2022 *)

is(n)=Set(factor(n)[,2]%2)==[1] \\ Charles R Greathouse IV, Feb 07 2017

More terms from Reinhard Zumkeller, Jun 23 2002

A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1

Offset: 1

Gerard P. Michon, Jul 05 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Gérard P. Michon, Multiplicative functions.

Cf. A005117, A076479, A162510, A162512, A002035, A072587, A036537, A162643, A162644, A162645, A046660, A008836, A307868.

A162511 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*(-1)^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after Reinhard Zumkeller *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,-(-1)^f[i]) \\ Charles R Greathouse IV, Mar 09 2015

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

from functools import reduce
from sympy import factorint
def A162511(n): return -1 if reduce(lambda a,b:~(a^b), factorint(n).values(),0)&1 else 1 # Chai Wah Wu, Jan 01 2023

Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
a(n) = 1 when n is a squarefree number (A005117).
From Reinhard Zumkeller, Jul 08 2009 (Start)
a(n) = (-1)^(A001222(n)-A001221(n)).
a(A162644(n)) = +1; a(A162645(n)) = -1. (End)
a(n) = A076479(n) * A008836(n). - R. J. Mathar, Mar 30 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A307868. - Amiram Eldar, Sep 18 2022
Dirichlet g.f.: Product_{p prime} ((p^s + 2)/(p^s + 1)). - Amiram Eldar, Oct 26 2023

A162645 Numbers m such that A162511(m) = -1.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 108, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Numbers n where A001222(n)-A001221(n) is odd. - Enrique Pérez Herrero, Jul 07 2012

This sequence has an asymptotic density (1 - A065472/zeta(2))/2 = 0.264159... (Mossinghoff and Trudgian, 2019). - Amiram Eldar, Jul 07 2020

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Complement of A162644.
Subsequence of A072587.
Cf. A001221, A001222, A065472, A162643, A162511.

Select[Range[10^2],OddQ[PrimeOmega[#]-PrimeNu[#]]&] (* Enrique Pérez Herrero, Jul 07 2012 *)

A340820 Decimal expansion of (1 + Product_{p prime} (1 - 2/(p*(p+1))))/2.

Original entry on oeis.org

7, 3, 5, 8, 4, 0, 3, 0, 6, 8, 0, 6, 4, 9, 8, 9, 3, 4, 0, 3, 7, 6, 1, 7, 8, 1, 6, 5, 4, 0, 2, 4, 1, 0, 4, 3, 7, 1, 2, 9, 6, 3, 1, 9, 1, 0, 0, 3, 4, 9, 3, 4, 4, 1, 8, 1, 7, 8, 6, 8, 6, 2, 7, 7, 0, 8, 8, 6, 6, 0, 5, 8, 3, 7, 9, 8, 4, 1, 3, 7, 2, 0, 4, 8, 1, 0, 5

Offset: 0

Amiram Eldar, Jan 22 2021

The asymptotic density of numbers k such that A046660(k) is even (A162644).

Detrey et al. (2016) calculated 1000 decimal digits of this constant.

			0.735840306806498934037617816540241043712963191003493...

Jérémie Detrey, Pierre-Jean Spaenlehauer and Paul Zimmermann, Computing the rho constant, 2016.
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019. See page 9.

Cf. A013661, A046660, A065472, A162644, A340818, A340819.

PARI
```
(prodeulerrat(1 - 2/(p*(p+1))) + 1)/2
```

Equals (1 + A065472/zeta(2))/2.

Equals lim_{n->oo} A340818(n)/A340819(n).

cp's OEIS Frontend

A340818 Numerators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

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A340819 Denominators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

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A036537 Numbers whose number of divisors is a power of 2.

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A002035 Numbers that contain primes to odd powers only.

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A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

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A162645 Numbers m such that A162511(m) = -1.

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A340820 Decimal expansion of (1 + Product_{p prime} (1 - 2/(p*(p+1))))/2.

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