A347195 -id:A347195 - cp's OEIS frontend

A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 16, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Aug 12 2018

a(n) is the denominator of certain rational valued sequences f(n), that have been defined as f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, dA034444 and A037445.
Many of the same observations as given in A046644 apply also here. Note that A011371 shares with A005187 the property that A011371(x+y) <= A011371(x) + A011371(y), with equivalence attained only when A004198(x,y) = 0, and also the property that A011371(2^(k+1)) = 1 + 2*A011371(2^k).
The following list gives such pairs num(n), b(n) for which b(n) is Dirichlet convolution of num(n)/a(n).
  Numerators   Dirichlet convolution of numerator(n)/a(n) yields
  -------      -----------
  A317933      A034444
  A317941      A037445
  A317940      A046644
Expansion of Dirichlet g.f. Product_{prime} 1/(1 - 2/p^s)^(1/2) is A046643/A317934. - Vaclav Kotesovec, May 08 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)
Wikipedia, Dirichlet convolution

Cf. A011371, A034444, A317940, A317941, A317946.
Cf. A317933, A317940, A317941 (numerator-sequences).
Cf. also A046644, A299150, A299152, A317832, A317932, A317926 (for denominator sequences of other similar constructions).
Cf. A046643, A167864, A347195.

A011371(n) = (n - hammingweight(n));
A317934(n) = factorback(apply(e -> 2^A011371(e),factor(n)[,2]));

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 07 2025

for(n=1, 100, print1(denominator(direuler(p=2, n, ((1+X)/(1-X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = 2^A317946(n).
a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d 1, where b is A034444, A037445 or A046644 for example.
Sum_{k=1..n} A046643(k)/a(k) ~ n * sqrt(A167864*log(n)/(Pi*log(2))) * (1 + (4*(gamma - 1) + 5*log(2) - 4*A347195)/(8*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 08 2025

A069205 a(n) = Sum_{k=1..n} 2^bigomega(k).

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 25, 29, 33, 35, 43, 45, 49, 53, 69, 71, 79, 81, 89, 93, 97, 99, 115, 119, 123, 131, 139, 141, 149, 151, 183, 187, 191, 195, 211, 213, 217, 221, 237, 239, 247, 249, 257, 265, 269, 271, 303, 307, 315, 319, 327, 329, 345, 349, 365, 369, 373

Offset: 1

Benoit Cloitre, Apr 14 2002

Partial sums of A061142. - Michel Marcus, Aug 08 2017

G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 53, exercise 5 (in the third edition 2015, p. 59, exercise 57).

Michel Marcus, Table of n, a(n) for n = 1..1000
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Cf. A061142, A167864, A347195, A350961.

Accumulate[2^PrimeOmega[Range[60]]] (* Harvey P. Dale, Aug 22 2011 *)

a(n) = sum(k=1, n, 2^bigomega(k)); \\ Michel Marcus, Aug 08 2017

Asymptotic formula: a(n) = 1/(8*log(2))*C*n*log(n)^2+O(n*log(n)) with C = A167864 = Product_{p primes > 2} (1+1/p/(p-2)) where the product is over all the primes p>2.
From Daniel Suteu, May 23 2020: (Start)
a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)) * floor(n/k).
a(n) = Sum_{k=1..n} A335073(floor(n/k)).
a(n) = 1 + Sum_{k=1..floor(log_2(n))} 2^k * pi_k(n), where pi_k(n) is the number of k-almost primes <= n. (End)
More precise asymptotics [Grosswald, 1956]: a(n) ~ A167864*n*log(n)*(log(n) - 2 - 4*A347195 + 4*gamma + 5*log(2))/(8*log(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 22 2021
Even more precise formula: a(n) ~ A167864 * n / (8*log(2)) * (log(n)^2 + (4*g + 5*log(2) - 2 - 4*A347195)*log(n) + 2 + 2*g^2 - 4*sg1 - 5*log(2) + 13*log(2)^2/6 + 2*g*(5*log(2) - 2) - 2*A347195*(5*log(2) - 2 + 4*g) + 4*A347195^2 + c), where c = Sum_{prime p > 2} (2*p * (2*p-3)* log(p)^2) / ((p-2)^2 * (p-1)^2) = 8.86809160013722347937514407919207620377461987744681170588044228288988578547..., g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 11 2022

A162510 Dirichlet inverse of A076479.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 32, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 16, 1, 2, 2, 4, 1, 1, 1, 4, 1

Offset: 1

Gerard P. Michon, Jul 05 2009

Apart from signs, this sequence is identical to A162512.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
Gérard P. Michon, Multiplicative functions.

Cf. A076479, A162511, A162512.
Cf. A000079, A001222, A005117, A034444, A046660, A061142.
Cf. A167864, A347195.

A162510 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*2^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := 2^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after R. J. Mathar *)

a(n)=my(f=factor(n)[,2]); 2^(vecsum(f)-#f) \\ Charles R Greathouse IV, Nov 02 2016

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [2**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

Multiplicative with a(p^e) = 2^(e-1) for any prime p and any positive exponent e.
a(n) = n/2 when n is a power of 2 (A000079).
a(n) = 1 when n is a squarefree number (A005117).
a(n) = 2^A046660(n) = A061142(n)/A034444(n). - R. J. Mathar, Nov 02 2016
a(n) = Sum_{d|n} mu(d) * 2^A001222(n/d). - Daniel Suteu, May 21 2020
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^omega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021
Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1/(1 - 2/p^s)). - Amiram Eldar, Sep 16 2023
Sum_{k=1..n} 1/a(k) = c * n + o(n), where c = Product_{p prime} (1 - 1/(p*(2*p-1))) = 0.74030830284678515949... (Jakimczuk, 2024, Theorem 2.4, p. 16). - Amiram Eldar, Mar 08 2024
From Vaclav Kotesovec, Mar 08 2024: (Start)
Dirichlet g.f.: zeta(s) * (1 + 1/(2^s*(2^s - 2))) * f(s), where f(s) = Product_{p prime, p>2} (1 + 1/(p^s*(p^s - 2))).
Sum_{k=1..n} a(k) ~ (f(1)*n / (4*log(2))) * (log(n) - 1 + gamma + 5*log(2)/2 + f'(1)/f(1)), where
f(1) = Product_{p prime, p>2} (1 + 1/(p*(p-2))) = A167864 = 1.51478012813749125771853381230067247330485921179389884042843306025133959...,
f'(1) = f(1) * Sum_{p prime, p>2} (-2*log(p)/((p-1)*(p-2))) = -2*f(1)*A347195 = -2.6035805486753944250682818932032862770113061830543948257159113584026980...
and gamma is the Euler-Mascheroni constant A001620. (End)

A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2p^(-2s)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 11, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 21, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 11, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 43, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 11, 11, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 21, 1, 3, 3, 9

Offset: 1

Ilya Gutkovskiy, Feb 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (2*10^8 terms)

Cf. A001045, A061142, A351219, A351346, A351348.

f[p_, e_] := (2^(e + 1) + (-1)^e)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X - 2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022

Multiplicative with a(p^e) = Jacobsthal(e+1).
From Vaclav Kotesovec, Feb 11 2022: (Start)
Let f(s) = Product_{prime p>2} (1 - 3/p^(2*s) + 2/p^(3*s))/(1 - 4/p^(2*s)), then
Sum_{k=1..n} a(k) ~ n*((2 * Pi^2 * log(n) + Pi^2 * (5*log(2) + 2*gamma - 2) + 24*zeta'(2))*f(1) + 2*Pi^2 * f'(1)) / (48*log(2)), where
f(1) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = A167864 = 1.5147801281374912577909192556494748924152701582862143953574842714849322098...,
f'(1) = -f(1) * Sum_{primes p > 2} 2*log(p) / (2 - 3*p + p^2) = -2*f(1)*A347195 = -2.603580548675394425068281893203286277011306183054394825715911358402698051... and gamma is the Euler-Mascheroni constant A001620. (End)

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A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

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A069205 a(n) = Sum_{k=1..n} 2^bigomega(k).

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A162510 Dirichlet inverse of A076479.

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A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2*p^(-2*s)).

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A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2p^(-2s)).