A167864 -id:A167864 - cp's OEIS frontend

A005597 Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).

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

Offset: 0

N. J. A. Sloane

C_2 = Product_{ p prime > 2} (p * (p-2) / (p-1)^2) is the 2-tuple case of the Hardy-Littlewood prime k-tuple constant (part of First H-L Conjecture): C_k = Product_{ p prime > k} (p^(k-1) * (p-k) / (p-1)^k).
Although C_2 is commonly called the twin prime constant, it is actually the prime 2-tuple constant (prime pair constant) which is relevant to prime pairs (p, p+2m), m >= 1.
The Hardy-Littlewood asymptotic conjecture for Pi_2m(n), the number of prime pairs (p, p+2m), m >= 1, with p <= n, claims that Pi_2m(n) ~ C_2(2m) * Li_2(n), where Li_2(n) = Integral_{2, n} (dx/log^2(x)) and C_2(2m) = 2 * C_2 * Product_{p prime > 2, p | m} (p-1)/(p-2), which gives: C_2(2) = 2 * C_2 as the prime pair (p, p+2) constant, C_2(4) = 2 * C_2 as the prime pair (p, p+4) constant, C_2(6) = 2* (2/1) * C_2 as the prime pair (p, p+6) constant, C_2(8) = 2 * C_2 as the prime pair (p, p+8) constant, C_2(10) = 2 * (4/3) * C_2 as the prime pair (p, p+10) constant, C_2(12) = 2 * (2/1) * C_2 as the prime pair (p, p+12) constant, C_2(14) = 2 * (6/5) * C_2 as the prime pair (p, p+14) constant, C_2(16) = 2 * C_2 as the prime pair (p, p+16) constant, ... and, for i >= 1, C_2(2^i) = 2 * C_2 as the prime pair (p, p+2^i) constant.
C_2 also occurs as part of other Hardy-Littlewood conjectures related to prime pairs, e.g., the Hardy-Littlewood conjecture concerning the distribution of the Sophie Germain primes (A156874) on primes p such that 2p+1 is also prime.
Another constant related to the twin primes is Viggo Brun's constant B (sometimes also called the twin primes Viggo Brun's constant B_2) A065421, where B_2 = Sum (1/p + 1/q) as (p,q) runs through the twin primes.
Reciprocal of the Selberg-Delange constant A167864. See A167864 for additional comments and references. - Jonathan Sondow, Nov 18 2009
C_2 = Product_{prime p>2} (p-2)p/(p-1)^2 is an analog for primes of Wallis' product 2/Pi = Product_{n=1 to oo} (2n-1)(2n+1)/(2n)^2. - Jonathan Sondow, Nov 18 2009
One can compute a cubic variant, product_{primes >2} (1-1/(p-1)^3) = 0.855392... = (2/3) * 0.6601618...* 1.943596... by multiplying this constant with 2/3 and A082695. - R. J. Mathar, Apr 03 2011
Cohen (1998, p. 7) referred to this number as the "twin prime and Goldbach constant" and noted that, conjecturally, the number of twin prime pairs (p,p+2) with p <= X tends to 2*C_2*X/log(X)^2 as X tends to infinity. - Artur Jasinski, Feb 01 2021

			0.6601618158468695739278121100145557784326233602847334133194484233354056423...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 84-93, 133.
R. K. Guy, Unsolved Problems in Number Theory, Section A8.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 194, 263-264.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..1001
Folkmar Bornemann, PRIMES Is in P: Breakthrough for "Everyman", Notices Amer. Math. Soc., Vol. 50, No. 5 (May 2003), p. 549.
Paul S. Bruckman, Problem H-576, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 39, No. 4 (2001), p. 379; General IZE, Solution to Problem H-576 by the proposer, ibid., Vol. 40, No. 4 (2002), pp. 383-384.
C. K. Caldwell, The Prime Glossary, twin prime constant.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Steven R. Finch, Mathematical Constants, Errata and Addenda, arXiv:2001.00578 [math.HO], 2020, Sec. 2.1.
Philippe Flajolet and Ilan Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
Daniel A. Goldston, Timothy Ngotiaoco and Julian Ziegler Hunts, The tail of the singular series for the prime pair and Goldbach problems, Functiones et Approximatio Commentarii Mathematici, Vol. 56, No. 1 (2017), pp. 117-141; arXiv preprint, arXiv:1409.2151 [math.NT], 2014.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant T_1^(2).
B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
G. Niklasch, Twin primes constant.
Simon Plouffe, The twin primes constant.
Simon Plouffe, Plouffe's Inverter, The twin primes constant.
Pascal Sebah, Numbers, constants and computation (gives 5000 digits).
Eric Weisstein's World of Mathematics, Twin Primes Constant.
Eric Weisstein's World of Mathematics, Twin Prime Conjecture.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.
Eric Weisstein's World of Mathematics, Prime Constellation.
John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.

Cf. A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271, A114907, A065418 (C_3), A167864, A000010, A008683.

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[C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 15 2012, after PARI *)
digits = 105; f[n_] := -2*(2^n-1)/(n+1); C2 = Exp[NSum[f[n]*(PrimeZetaP[n+1] - 1/2^(n+1)), {n, 1, Infinity}, NSumTerms -> 5 digits, WorkingPrecision -> 5 digits]]; RealDigits[C2, 10, digits][[1]] (* Jean-François Alcover, Apr 16 2016, updated Apr 24 2018 *)

\p1000; 175/256*prod(k=2,500,(zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k,d,moebius(d)*2^(k/d))/k))

prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 12 2021

Equals Product_{k>=2} (zeta(k)*(1-1/2^k))^(-Sum_{d|k} mu(d)*2^(k/d)/k). - Benoit Cloitre, Aug 06 2003
Equals 1/A167864. - Jonathan Sondow, Nov 18 2009
Equals Sum_{k>=1} mu(2*k-1)/phi(2*k-1)^2, where mu is the Möbius function (A008683) and phi is the Euler totient function (A000010) (Bruckman, 2001). - Amiram Eldar, Jan 14 2022

More terms from Vladeta Jovovic, Nov 08 2001
Commented and edited by Daniel Forgues, Jul 28 2009, Aug 04 2009, Aug 12 2009
PARI code removed by D. S. McNeil, Dec 26 2010

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

Original entry on oeis.org

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)

A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 1, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 1, 20, 16, 8, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 2, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 12, 1, 16, 20, 66, 16, 44, 8, 70, 2, 72, 36

Offset: 1

Artur Jasinski, Feb 05 2025

This sequence is similar to A173557 but differences occurs for indices n=35,65,70,...
Coefficients J(n)=a(n)/A307410(n) occurs in many formulas on density of primes with gap 2*n.
Sylvester was the first who uses these coefficients at 1871.

			1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3, 10/9, 2, 12/11, ...
a(35) = 8 because 35 = 5 * 7 and then product is ((5-1)/(5-2))*((7-1)/(7-2)) = 8/5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70 (e.g. formula 4.11 p.32).
J. J. Sylvester, On the partition of an even number into two primes, Proc. London Math. Soc., ser. I, vol.4 (1871), pp. 4-6 (Math. Papers, vol. 2, pp. 709-711).
Marek Wolf, Applications of statistical mechanics in number theory, Physica A 274 (1999) 149-157 (formula (9) p. 154).

Cf. A167864, A173557, A305444, A307410 (denominators).

j = {}; Do[prod = 1; Do[If[PrimeQ[n] && IntegerQ[d/n], prod = prod (n - 1)/(n - 2)], {n, 3, d}]; AppendTo[j, prod], {d, 1, 74}]; Numerator[j]
f[p_, e_] := If[p == 2, 1, (p-1)/(p-2)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 03 2025 *)

a(n) = my(f=factor(n)[,1]); numerator(prod(k=1, #f, if ((p=f[k])>2, (p-1)/(p-2), 1))); \\ Michel Marcus, Feb 05 2025

a(n) = numerator(A173557(n)/A305444(n)).
a(p^n) = p - 1 for prime p.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A307410(k) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = 1.51478012... (A167864). - Amiram Eldar, Mar 03 2025

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/(zeta(2*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

			1.8107814761215629522431259...

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Mathematica
```
RealDigits[N[5005/2764,105]][[1]]
```

default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.25(3617945007235890014471780028943560057887120115774240231548480463096960
9261939218523878437047756874095513748191027496382054992764109985528219
9710564399421128798842257597684515195369030390738060781476121562952243
12590448625180897250).

			1.25361794500723589001447178...

Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340065.

Mathematica
```
RealDigits[N[3465/2764, 105]][[1]]
```

default(realprecision, 105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

Equals 3465/2764 = 3^2*5*7*11/(2^2*691).
Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.
Equals (9/13)*A340065.

cp's OEIS Frontend

A005597 Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A162510 Dirichlet inverse of A076479.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Views

Author

Keywords

Comments

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