A299822 -id:A299822 - cp's OEIS frontend

A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

Also decimal expansion of the probability that an integer triple (x, y, z) is pairwise coprime. - Charles R Greathouse IV, Nov 14 2011

The probability that 2 numbers chosen at random are coprime, and both squarefree (Delange, 1969). - Amiram Eldar, Aug 04 2020

			0.2867474284344787341078927127898384...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.

Juan Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4
Tim Browning, The divisor problem for binary cubic forms, Journal de théorie des nombres de Bordeaux, Vol. 23, No. 3 (2011), pp. 579-602; arXiv preprint, arXiv:1006.3476 [math.NT], 2010.
Hubert Delange, On some sets of pairs of positive integers, Journal of Number Theory, Vol. 1, No. 3 (1969), pp. 261-279. See p. 277.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 181.
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [cached copy]
László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quart., Vol. 40 (2002), pp. 13-18.
László Tóth, Another generalization of Euler's arithmetic function and of Menon's identity, arXiv:2006.12438 [math.NT], 2020. See p. 3.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A065472, A069201, A069212, A074816, A074823, A078073, A098198, A227929, A299822.

digits = 100; NSum[-(2+(-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 11 2016 *)

prodeulerrat(1 - (3*p-2)/(p^3)) \\ Amiram Eldar, Mar 17 2021

Equals Prod_{p prime} (1 - 1/p)^2*(1 + 2/p). - Michel Marcus, Apr 16 2016
The constant c in Sum_{k<=x} mu(k)^2 * 2^omega(k) = c * x * log(x) + O(x), where mu is A008683 and omega is A001221, and in Sum_{k<=x} 3^omega(k) = (1/2) * c * x * log(x)^2 + O(x*log(x)) (see Tenenbaum, 2015). - Amiram Eldar, May 24 2020
Equals A065472 * A227929 =  A065472 / A098198. - Amiram Eldar, Aug 04 2020

Name corrected by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 03 2003

More digits from Vaclav Kotesovec, Dec 19 2019

A009262 a(n) = lcm(n, phi(n)).

Original entry on oeis.org

1, 2, 6, 4, 20, 6, 42, 8, 18, 20, 110, 12, 156, 42, 120, 16, 272, 18, 342, 40, 84, 110, 506, 24, 100, 156, 54, 84, 812, 120, 930, 32, 660, 272, 840, 36, 1332, 342, 312, 80, 1640, 84, 1806, 220, 360, 506, 2162, 48, 294, 100, 1632, 312, 2756, 54, 440, 168, 684, 812, 3422

Offset: 1

David W. Wilson

nonn

This is a divisibility sequence: if n divides m, a(n) divides a(m). - Franklin T. Adams-Watters, Mar 30 2010
a(n) = n iff n is in A007694.
a(n) is a divisor of A299822(n). It is a proper divisor iff n is in A069209. - Max Alekseyev, Oct 11 2024

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index to divisibility sequences

Cf. A000010, A007694, A007947, A052106, A069209, A076512, A109395, A174824.

with(numtheory); A009262:=n->lcm(n, phi(n)); seq(A009262(n), n=1..100); # Wesley Ivan Hurt, Jan 29 2014

Table[LCM[n, EulerPhi[n]], {n, 100}] (* Wesley Ivan Hurt, Jan 29 2014 *)

a(n)=lcm(eulerphi(n), n) \\ Charles R Greathouse IV, May 29 2016

a(n) = A000010(n) * A109395(n) = n * A076512(n) = A299822(n) / gcd(A007947(n),phi(A007947(n))). - Max Alekseyev, Oct 11 2024

A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

Original entry on oeis.org

1, 4, 6, 6, 10, 24, 14, 8, 9, 40, 22, 36, 26, 56, 60, 10, 34, 36, 38, 60, 84, 88, 46, 48, 15, 104, 12, 84, 58, 240, 62, 12, 132, 136, 140, 54, 74, 152, 156, 80, 82, 336, 86, 132, 90, 184, 94, 60, 21, 60, 204, 156, 106, 48, 220, 112, 228, 232, 118, 360, 122, 248, 126, 14, 260

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A001221, A005117, A007947, A016754 (numbers n such that a(n) is odd), A034444, A038040, A064549, A299822, A304407, A304408, A304410 (fixed points), A304411, A304412.

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]

a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A007947(n).
a(p^k) = p*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 2/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, Jun 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-1) + 2/p^(s-1) + 1/p^(2*s) - 2/p^s) * ((p^s - p)/(p^s - 1))^2.
Dirichlet g.f.: zeta(s-1)^2 * f(s).
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 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,
f'(2) = f(2) * Sum_{p prime} 2*(3*p^4-3*p^2+1) * log(p) / ((p-1)*(p+1)*(p^4+2*p^3-2*p^2-p+1)) = f(2) * 2.2612432627709318567813765271568350301741329636853...
and gamma is the Euler-Mascheroni constant A001620. (End)

A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2.

Original entry on oeis.org

1, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 30, 8, 136, 9, 171, 20, 63, 55, 253, 12, 50, 78, 27, 42, 406, 30, 465, 16, 165, 136, 210, 18, 666, 171, 234, 40, 820, 63, 903, 110, 90, 253, 1081, 24, 147, 50, 408, 156, 1378, 27, 550, 84, 513, 406, 1711, 60, 1830, 465, 189

Offset: 1

Benoit Cloitre, Feb 12 2004

a(2n+1) is the conjectured value of the length of period of sequence of Genocchi number of first kind read modulo (2n + 1) (cf. A001469).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A023900, A034444. - R. J. Mathar, Feb 08 2011

Cf. A299822, A000010, A007947, A001221.

A023900 := proc(n) add( d*numtheory[mobius](d),d=numtheory[divisors](n)) ; end proc:
A001221 := proc(n) nops(numtheory[factorset](n)) ; end proc:
A076479 := proc(n) (-1)^A001221(n) ; end proc:
A034444 := proc(n) 2^A001221(n) ;end proc:
A090780 := proc(n) n/A076479(n)/A034444(n) *A023900(n); end proc:
seq(A090780(n),n=1..20) ; # R. J. Mathar, Apr 14 2011

a[n_] := Module[{f, p, e}, fun[p_, e_] := (p - 1)*p^e/2;
If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]]; Array[a, 50] (* Amiram Eldar, Nov 23 2018 *)

a(n) = my(f=factor(n)[,1]); n*prod(k=1, #f, (f[k]-1)/2); \\ Michel Marcus, May 26 2019

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]/2) \\ Jianing Song, Aug 11 2023

a(n) = (n/(-2)^omega(n))*(Sum_{d|n} d*mu(d)) = n*A023900(n)/(A076479(n)*A034444(n)).
a(n) = n*A173557(n)/2. - R. J. Mathar, Apr 14 2011
From Jianing Song, Nov 22 2018: (Start)
Multiplicative with a(p^e) = (p - 1)*p^e/2 = A000217(p-1)*p^(e-1).
a(n) = A299822(n)/2^A001221(n).
a(prime(n)) = A034953(n).
a(n) is odd if and only if n = A004614(k) or 2*A004614(k). (End)
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 2/(p-1)^2) = 5.72671092223951683002237367406848393189560038246828458038126468772919585... - Vaclav Kotesovec, Sep 20 2020
From Jianing Song, Aug 11 2023: (Start)
a(n) = phi(n) * Product_{p|n, p prime} (p/2), where phi = A000010.
Equals A000010(n)*A007947(n)/2^A001221(n). (End)

A088847 a(n) = sigma(A087979(n)) / phi(A087979(n)).

Original entry on oeis.org

1, 1, 3, 4, 4, 6, 6, 8, 9, 8, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 20, 22, 24, 25, 24, 27, 28, 28, 30, 30, 32, 33, 32, 35, 36, 36, 36, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 56, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Labos Elemer, Nov 17 2003

nonn

Note that A087979(n) is a balanced number (A020492), hence this sequence is well-defined. - Max Alekseyev, Oct 10 2024

For all n, a(n) <= n and a(n) divides A299822(n). - Max Alekseyev, Oct 11 2024

			While A088830 includes special balanced numbers, A087979 does not include per definition. Nevertheless, it seems that A087979 consists only of balanced numbers. This is provable at least for special cases.

Cf. A000010, A000203, A087979, A020492, A088830, A299822.

a(24)-a(35) from Amiram Eldar, Dec 05 2019 (calculated from the data at A087979)

Terms a(36) onward from Max Alekseyev, Oct 10 2024

cp's OEIS Frontend

A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

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A009262 a(n) = lcm(n, phi(n)).

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A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

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A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2.

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A088847 a(n) = sigma(A087979(n)) / phi(A087979(n)).

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