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A176345 Sum of gcd(k,n) from k = 1 to n over "regular" integers only (an integer k is regular if there is an x such that k^2 x == k (mod n)) 2
1, 3, 5, 6, 9, 15, 13, 12, 15, 27, 21, 30, 25, 39, 45, 24, 33, 45, 37, 54, 65, 63, 45, 60, 45, 75, 45, 78, 57, 135, 61, 48, 105, 99, 117, 90, 73, 111, 125, 108, 81, 195, 85, 126, 135, 135, 93, 120, 91, 135, 165, 150, 105, 135, 189, 156, 185, 171, 117, 270, 121, 183, 195 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

It is also the product of n and (2-1/p), taken over all primes p dividing n.

LINKS

Daniel Suteu, Table of n, a(n) for n = 1..10000

S. Chen and W. Zhai, Reciprocals of the Gcd-Sum Functions, J. Int. Seq. 14 (2011) # 11.8.3.

J.-M. De Koninck and I. Katai, Some remarks on a paper of L. Toth, JIS 13 (2010) 10.1.2.

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Laszlo Toth, A Gcd-Sum Function Over Regular Integers Modulo n, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.5.

Laszlo Toth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

D. Zhang and W. Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq. 13 (2010), 10.4.7.

D. Zhang and W. Zhai, Mean Values of a Class of Arithmetical Functions, J. Int. Seq. 14 (2011) #11.6.5.

FORMULA

Multiplicative with a(p^e) = 2*p^e-p^(e-1).

Dirichlet g.f.: zeta(s-1)*product_p (1+p^(1-s)-p^(-s)). Dirichlet convolution of the series of absolute values of A097945 with A000027. - R. J. Mathar, Jun 11 2011

From Daniel Suteu, Jun 27 2018: (Start)

a(n) = Sum_{d|n} d * phi(n/d) * mu(n/d)^2.

a(n) = Sum_{d|n, gcd(n/d, d) = 1} d * phi(n/d). (End)

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2*gcd(n,k).

a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*n/gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

From Vaclav Kotesovec, Aug 20 2021: (Start)

Dirichlet g.f.: zeta(s-1)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).

Let f(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)), then Sum_{k=1..n} a(k) ~ n^2 * (f(2)*(log(n)/2 + gamma - 1/4) + f'(2)/2), where f(2) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094..., f'(2) = f(2) * Sum_{primes p} log(p) * (3*p - 2) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407300858... and gamma is the Euler-Mascheroni constant A001620. (End)

EXAMPLE

For n = 8, the regular integers mod 8 are 1,3,5,7,8, so the sum of gcd's of 8 with these numbers is 12.

MAPLE

A176345 := proc(n)

    n*mul(2-1/p, p=numtheory[factorset](n)) ;

end proc:

seq(A176345(n), n=1..40) ; # R. J. Mathar, Sep 13 2016

MATHEMATICA

f[p_, e_] := 2*p^e - p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

PROG

(PARI) isregg(k, n) = {g = gcd(k, n); if ((n % g == 0) && (gcd(g, n/g) == 1), return(g), return(0)); }

a(n) = sum(k=1, n, isregg(k, n)) \\ Michel Marcus, May 25 2013

(PARI) a(n) = sumdiv(n, d, d * eulerphi(n/d) * moebius(n/d)^2); \\ Daniel Suteu, Jun 27 2018

(PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, 2*f[k, 1]^f[k, 2] - f[k, 1]^(f[k, 2]-1)); \\ Daniel Suteu, Jun 27 2018

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X^2 - p^2*X^2 - X)/(1-p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

CROSSREFS

Cf. A143869.

Sequence in context: A294909 A070111 A070117 * A101139 A309140 A102606

Adjacent sequences:  A176342 A176343 A176344 * A176346 A176347 A176348

KEYWORD

nonn,mult,easy

AUTHOR

Jeffrey Shallit, Apr 15 2010

STATUS

approved

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Last modified October 27 13:19 EDT 2021. Contains 348276 sequences. (Running on oeis4.)