A069205 -id:A069205 - cp's OEIS frontend

A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 32, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 32, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 64, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 32, 16, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4

Offset: 1

Henry Bottomley, May 29 2001

The inverse Möbius transform of A162510. - R. J. Mathar, Feb 09 2011

			a(100)=16 since 100=2*2*5*5 and so a(100)=2*2*2*2.

R. Zumkeller, Table of n, a(n) for n = 1..10005
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011-2012. See eq. (2.12).
Index entries for sequences computed from exponents in factorization of n

Cf. A000079, A001222, A001316, A034444, A069205 (partial sums), A123667, A124508, A156552.

with(numtheory): seq(2^bigomega(n),n=1..95);

Table[2^PrimeOmega[n], {n, 1, 95}] (* Jean-François Alcover, Jun 08 2013 *)

a(n)=direuler(p=1,n,1/(1-2*X))[n] /* Ralf Stephan, Mar 28 2015 */

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

a(n) = Sum_{d divides n} 2^(bigomega(d)-omega(d)) = Sum_{d divides n} 2^(A001222(d) - A001221(d)). - Benoit Cloitre, Apr 30 2002
a(n) = A000079(A001222(n)), i.e., a(n)=2^bigomega(n). - Emeric Deutsch, Feb 13 2005
Totally multiplicative with a(p) = 2. - Franklin T. Adams-Watters, Oct 04 2006
Dirichlet g.f.: Product_{p prime} 1/(1-2*p^(-s)). - Ralf Stephan, Mar 28 2015
a(n) = A001316(A156552(n)). - Antti Karttunen, May 29 2017
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2). - Vaclav Kotesovec, Mar 14 2023

A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Nov 13 2009, Nov 17 2009

Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.
Numerators of partial products are A062271. Denominators are A062270.
An analog for primes of Wallis' product pi/2 = Product_{n >=1} (2n)^2/(2n-1)(2n+1), because A167864 = Product_{prime p>2} (p-1)^2/(p-2)p.
Grosswald (see links) proves that Sum_{k<=x} 2^Omega(k) ~ (1/(8*log(2))) * c * x * (log(x))^2 + O(x * log(x)) where c is this constant. - Amiram Eldar, Jun 06 2020
The asymptotic density of numbers m with A046660(m) = Omega(m) - omega(m) = k is asymptotically ~ c/2^(k+2) as k -> oo, where c is this constant (Rényi, 1955). - Amiram Eldar, Aug 08 2020
Named after the Norwegian mathematician Atle Selberg (1917-2007) and the French mathematician Hubert Delange (1914-2003). - Amiram Eldar, Jun 20 2021

			Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.5147801281374912577909192556...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc., Vol. 18, No. 1 (1954), pp. 83-87.
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.

Michel Balazard, Hubert Delange and Jean-Louis Nicolas, Sur le nombre de facteurs premiers des entiers, C. R. Acad. Sci., Paris, Ser. I, Vol. 306 (1988), pp. 511-514. [From Jonathan Sondow, Nov 17 2009]
Hubert Delange, Sur des formules de Atle Selberg, Acta Arith., Vol. 19 (1971), pp. 105-146.
Steven R. Finch, Mathematical Constants, Errata and Addenda, Sec. 2.1.
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
Jean-Louis Nicolas, Sur la distribution des nombres entiers ayant une quantite fixée de facteurs premiers, Acta Arith., Vol. 44 (1984), pp. 191-200.
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.

Cf. A005597.

Cf. A001222 (Omega), A046660, A061142 (2^Omega), A069205 (partial sums of 2^Omega).

s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, 160}]; RealDigits[1/C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 30 2012, after Pari program in A005597 *)
$MaxExtraPrecision = 300; digits = 105; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{3, -2}, {2, 6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 19 2016 *)

prodeulerrat((1 + 1/(p*(p-2))),,3) \\ Hugo Pfoertner, Aug 08 2020

Equals 1/A005597.

Equals Product_{prime p>2} (p-1)^2/(p-2)p = (2^2/1*3)(4^2/3*5)(6^2/5*7)(10^2/9*11) ....

A347195 Decimal expansion of Sum_{primes p > 2} log(p) / ((p-2)*(p-1)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 22 2021

Constant is related to the asymptotics of A069205.

			0.8593922313585686897187145141861232817699609176983112114741634265903839649...

Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44. [constant C3]

Cf. A069205, A351347.

ratfun = 1/((p-2)*(p-1)); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*(Zeta'[power]/Zeta[power] + Log[2]/(2^power - 1)) /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 2, m}] + zetas, 110]], {m, 2000, 10000, 2000}]

A350961 a(n) = Sum_{k=1..n} 3^Omega(k).

Original entry on oeis.org

1, 4, 7, 16, 19, 28, 31, 58, 67, 76, 79, 106, 109, 118, 127, 208, 211, 238, 241, 268, 277, 286, 289, 370, 379, 388, 415, 442, 445, 472, 475, 718, 727, 736, 745, 826, 829, 838, 847, 928, 931, 958, 961, 988, 1015, 1024, 1027, 1270, 1279, 1306, 1315, 1342, 1345, 1426, 1435, 1516, 1525, 1534, 1537, 1618

Offset: 1

N. J. A. Sloane, Feb 06 2022

Tenenbaum, G. (2015). Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. See page 59.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A001222 (Omega), A069205, A069212. Partial sums of A165824.

Accumulate[3^PrimeOmega[Range[100]]] (* Vaclav Kotesovec, Feb 16 2022 *)

from sympy.ntheory.factor_ import primeomega
def A350961(n): return sum(3**primeomega(m) for m in range(1,n+1)) # Chai Wah Wu, Sep 07 2023

A335073 a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 12, 14, 15, 16, 18, 19, 20, 21, 29, 30, 32, 33, 35, 36, 37, 38, 42, 44, 45, 49, 51, 52, 53, 54, 70, 71, 72, 73, 77, 78, 79, 80, 84, 85, 86, 87, 89, 91, 92, 93, 101, 103, 105, 106, 108, 109, 113, 114, 118, 119, 120, 121, 123, 124, 125, 127

Offset: 1

Daniel Suteu, May 22 2020

nonn

Partial sums of A162510.

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

Cf. A001221, A001222, A046660, A061142, A069205, A162510.

a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
      2^(bigomega(n)-nops(factorset(n)))+a(n-1))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, May 22 2020

Accumulate[Table[2^(PrimeOmega[n]-PrimeNu[n]),{n,70}]] (* Harvey P. Dale, Aug 14 2020 *)

a(n) = sum(k=1, n, 2^(bigomega(k) - omega(k)));

a(n) = Sum_{k=1..n} A008683(k) * A069205(floor(n/k)).

a(n) = Sum_{k=1..n} A061142(k) * A002321(floor(n/k)).

A166632 Totally multiplicative sequence with a(p) = 2*(p-1) for prime p.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 12, 8, 16, 16, 20, 16, 24, 24, 32, 16, 32, 32, 36, 32, 48, 40, 44, 32, 64, 48, 64, 48, 56, 64, 60, 32, 80, 64, 96, 64, 72, 72, 96, 64, 80, 96, 84, 80, 128, 88, 92, 64, 144, 128

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.

Cf. A001222, A001620, A003958, A061142, A069205.

f:= proc(n) local f;
  mul((2*(f[1]-1))^f[2], f = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, May 19 2016

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*2^(PrimeOmega[m]), {m, 1, 100}](* G. C. Greubel, May 19 2016, based on A003958 *)
f[p_, e_] := (2*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*(p-1)*X))[n], ", ")) \\ Vaclav Kotesovec, Mar 08 2023

Multiplicative with a(p^e) = (2*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-1))^e(k).
a(n) = A061142(n) * A003958(n) = 2^bigomega(n) * A003958(n) = 2^A001222(n) * A003958(n).
Dirichlet g.f.: Product_{p prime} 1/(1 - 2*(p-1)*p^(-s)). - Robert Israel, May 19 2016
From Vaclav Kotesovec, Mar 08 2023: (Start)
Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - (2 - p^(2-s))/(p^s-2*p+2)).
Let f(s) = Product_{p prime} (1 - (2 - p^(2-s)) / (p^s - 2*p + 2)).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - 2/(p^2 - 2*p + 2)) = 0.353804459718477500968617797456682002952375753701841967763205003892191...,
f'(2) = f(2) * Sum_{p prime} 2*log(p) / ((p-1) * (p^2 - 2*p + 2)) = 0.350193097012820163529213089258238034020398107720137317340667886409682...
 and gamma is the Euler-Mascheroni constant A001620. (End)

A375286 a(n) = f(1) + f(2) + ... + f(n), where f(n) = (-2)^Omega(n) = A165872(n).

Original entry on oeis.org

1, -1, -3, 1, -1, 3, 1, -7, -3, 1, -1, -9, -11, -7, -3, 13, 11, 3, 1, -7, -3, 1, -1, 15, 19, 23, 15, 7, 5, -3, -5, -37, -33, -29, -25, -9, -11, -7, -3, 13, 11, 3, 1, -7, -15, -11, -13, -45, -41, -49, -45, -53, -55, -39, -35, -19, -15, -11, -13, 3, 1, 5, -3, 61

Offset: 1

Charles R Greathouse IV, Aug 09 2024

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Daniel R. Johnston, Nicol Leong, and Sebastian Tudzi, New bounds and progress towards a conjecture on the summatory function of (-2)^Omega(n), arXiv:2408.04143 [math.NT], 2024.
Michael J. Mossinghoff and Timothy S. Trudgian, Oscillations in weighted arithmetic sums, arXiv:2007.14537 [math.NT], 2020.
Zhi-Wei Sun, On a pair of zeta functions, arXiv:1204.6689 [math.NT], 2012.

Partial sums of A165872.

Cf. A001222, A002321, A212496, A061142, A069205.

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+(-2)^numtheory[bigomega](n))
    end:
seq(a(n), n=1..64);  # Alois P. Heinz, Apr 25 2025

PARI
```
s=0; vector(60,n,s+=(-2)^bigomega(n))
```

Johnston, Leong, & Tudzi prove that |a(n)| < 2260n. Sun conjectures that |a(n)| < n for n >= 3078. Mossinghoff & Trudgian verify this to 2.5 * 10^14.

Because of powers of two, |a(n)| >= n/2 infinitely often.

A378482 Decimal expansion of 1/(8log(2)A005597), where A005597 is the twin prime constant C_2.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 28 2024

			0.27317072236263839747106601431655151479129736936570...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.5.1, p. 111.
Gérald 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).

Paul T. Bateman, Proof of a conjecture of Grosswald, Duke Mathematical Journal, Vol. 25. No. 1 (1958), pp. 67-72.
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.

Cf. A002162, A005597, A069205.

1/(8*log(2)*prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Nov 29 2024

Equals lim_{n->oo} (1/(n*log(n)^2)) * A069205(n). - Amiram Eldar, Feb 15 2025

cp's OEIS Frontend

A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.

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A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

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A347195 Decimal expansion of Sum_{primes p > 2} log(p) / ((p-2)*(p-1)).

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A350961 a(n) = Sum_{k=1..n} 3^Omega(k).

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A335073 a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)).

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A166632 Totally multiplicative sequence with a(p) = 2*(p-1) for prime p.

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A375286 a(n) = f(1) + f(2) + ... + f(n), where f(n) = (-2)^Omega(n) = A165872(n).

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A378482 Decimal expansion of 1/(8log(2)A005597), where A005597 is the twin prime constant C_2.