A018804 Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

1, 3, 5, 8, 9, 15, 13, 20, 21, 27, 21, 40, 25, 39, 45, 48, 33, 63, 37, 72, 65, 63, 45, 100, 65, 75, 81, 104, 57, 135, 61, 112, 105, 99, 117, 168, 73, 111, 125, 180, 81, 195, 85, 168, 189, 135, 93, 240, 133, 195, 165, 200, 105, 243, 189, 260, 185, 171, 117, 360

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• mult

Comments

a(n) is the number of times the number 1 appears in the character table of the cyclic group C_n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 02 2001

a(n) is the number of ways to express all fractions f/g whereby each product (f/g)*n is a natural number between 1 and n (using fractions of the form f/g with 1 <= f,g <= n). For example, for n=4 there are 8 such fractions: 1/1, 1/2, 2/2, 3/3, 1/4, 2/4, 3/4 and 4/4. - Ron Lalonde (ronronronlalonde(AT)hotmail.com), Oct 03 2002

Number of non-congruent solutions to xy == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Conjecture: n>1 divides a(n)+1 iff n is prime. - Thomas Ordowski, Oct 22 2014

The above conjecture is false, with counterexample given by n = 3*37*43*42307*116341 and a(n)+1 = 26*n. - Varun Vejalla, Jun 19 2025

a(n) is the number of 0's in the multiplication table Z/nZ (cf. A000010 for number of 1's). - Eric Desbiaux, Jun 11 2015

{a(n)} == 1, 3, 1, 0, 1, 3, 1, 0, ... (mod 4). - Isaac Saffold, Dec 30 2017

Since a(p^e) = p^(e-1)*((p-1)e+p) it follows a(p) = 2p-1 and therefore p divides a(p)+1. - Ruediger Jehn, Jun 23 2022

Examples

			G.f. = x + 3*x^2 + 5*x^3 + 8*x^4 + 9*x^5 + 15*x^6 + 13*x^7 + 20*x^8 + ...
		

References

• S. S. Pillai, On an arithmetic function, J. Annamalai University 2 (1933), pp. 243-248.
• J. Sándor, A generalized Pillai function, Octogon Mathematical Magazine Vol. 9, No. 2 (2001), 746-748.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 2000 terms from T. D. Noe)
• U. Abel, W. Awan, and V. Kushnirevych, A Generalization of the Gcd-Sum Function, J. Int. Seq. 16 (2013) #13.6.7.
• Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
• Olivier Bordelles, The Composition of the gcd and Certain Arithmetic Functions, J. Int. Seq. 13 (2010) #10.7.1.
• Olivier Bordelles, An Asymptotic Formula for Short Sums of Gcd-Sum Functions, Journal of Integer Sequences, Vol. 15 (2012), #12.6.8.
• Olivier Bordelles, A Multidimensional Cesáro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.
• Kevin A. Broughan, The gcd-sum function, Journal of Integer Sequences 4 (2001), Article 01.2.2, 19 pp.
• J.-M. De Koninck and I. Kátai, Some remarks on a paper of L. Toth, JIS 13 (2010) 10.1.2.
• Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
• Soichi Ikeda and Kaneaki Matsuoka, On the Lcm-Sum Function, Journal of Integer Sequences, Vol. 17 (2014), Article 14.1.7.
• Mathematical Reflections, Solution to Problem O364, Issue 2, 2016, p 24. [Cached copy at Wayback Machine]
• Taylor McAdam, Almost-primes in horospherical flows on the space of lattices, arXiv:1802.08764 [math.DS], 2018.
• Taylor McAdam, Almost-prime times in horospherical flows on the space of lattices, Journal of Modern Dynamics (2019) Vol. 15, 277-327.
• J. Ransford, A Sum of Gcd’s over Friable Numbers, JIS vol 19 (2016) # 16.3.2
• Jeffrey Shallit, Problem E 2821, American Mathematical Monthly 87 (1980), 220.
• Jeffrey Shallit, Solution, American Mathematical Monthly, 88 (1981), 444-445.
• László Tóth, A gcd-sum function over regular integers modulo n, JIS 12 (2009) 09.2.5.
• László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2.
• László Tóth, A survey of gcd-sum functions, J. Integer Sequences 13 (2010), Article 10.8.1, 23 pp.
• László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
• 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.
• D. Zhang and W. Zhai, On an Open Problem of Tóth, J. Int. Seq. 16 (2013) #13.6.5.

Crossrefs

Column 1 of A343510 and A343516.

Cf. A080997, A080998 for rankings of the positive integers in terms of centrality, defined to be the average fraction of an integer that it shares with the other integers as a gcd, or A018804(n)/n^2, also A080999, a permutation of this sequence (A080999(n) = A018804(A080997(n))).

Cf. A185210, A010766, A054523, A127468, A050873, A078430, A095026, A227507, A000010, A344372, A272718 (partial sums), A368736 - A368741.

Programs

Haskell

a018804 n = sum $ map (gcd n) [1..n]  -- Reinhard Zumkeller, Jul 16 2012

Magma

[&+[Gcd(n,k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Nov 14 2019

Maple

a:=n->sum(igcd(n,j),j=1..n): seq(a(n), n=1..60); # Zerinvary Lajos, Nov 05 2006

Mathematica

f[n_] := Block[{d = Divisors[n]}, Sum[ d*EulerPhi[n/d], {d, d}]]; Table[f[n], {n, 60}] (* Robert G. Wilson v, Mar 20 2012 *)
a[ n_] := If[ n < 1, 0, n Sum[ EulerPhi[d] / d, {d, Divisors@n}]]; (* Michael Somos, Jan 07 2017 *)
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jul 19 2019 *)

PARI

{a(n) = direuler(p=2, n, (1 - X) / (1 - p*X)^2)[n]}; /* Michael Somos, May 31 2000 */

PARI

a(n)={ my(ct=0); for(i=0,n-1,for(j=0,n-1, ct+=(Mod(i*j,n)==0) ) ); ct; } \\ Joerg Arndt, Aug 03 2013

PARI

a(n)=my(f=factor(n)); prod(i=1,#f~,(f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]) \\ Charles R Greathouse IV, Oct 28 2014

PARI

a(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ Michel Marcus, Jan 07 2017

Python

from sympy.ntheory import totient, divisors
print([sum(n*totient(d)//d for d in divisors(n)) for n in range(1, 101)]) # Indranil Ghosh, Apr 04 2017

Python

from sympy import factorint
from math import prod
def A018804(n): return prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()) # Chai Wah Wu, Nov 29 2021

Formula

a(n) = Sum_{d|n} d*phi(n/d), where phi(n) is Euler totient function (cf. A000010). - Vladeta Jovovic, Apr 04 2001

Multiplicative; for prime p, a(p^e) = p^(e-1)*((p-1)e+p).

Dirichlet g.f.: zeta(s-1)^2/zeta(s).

a(n) = Sum_{d|n} d*tau(d)*mu(n/d). - Benoit Cloitre, Oct 23 2003

Equals A054523 * [1,2,3,...]. Equals row sums of triangle A010766. - Gary W. Adamson, May 20 2007

Equals inverse Mobius transform of A029935 = A054525 * (1, 2, 4, 5, 8, 8, 12, 12, ...). - Gary W. Adamson, Aug 02 2008, corrected Feb 07 2023

Equals row sums of triangle A127478. - Gary W. Adamson, Aug 03 2008

G.f.: Sum_{k>=1} phi(k)*x^k/(1 - x^k)^2, where phi(k) is the Euler totient function. - Ilya Gutkovskiy, Jan 02 2017

a(n) = Sum_{a = 1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1, for n > 1. The sum is over a,b,c such that n*c - a*b = 0. - Benedict W. J. Irwin, Apr 04 2017

Proof: Let gcd(a, n) = g and x = n/g. Define B = {x, 2*x, ..., g*x}; then for all b in B there exists a number c such that a*b = n*c. Since the set B has g elements it follows that Sum_{b=1..n} Sum_{c=1..n} 1 >= g = gcd(a, n) and therefore Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 >= Sum_{a=1..n} gcd(a, n). On the other hand, for all b not in B there is no number c <= n such that a*b = n*c and hence Sum_{b = 1..n} Sum_{c = 1..n} 1 = g. Therefore Sum_{a=1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1 = a(n). - Ruediger Jehn, Jun 23 2022

a(2*n) = a(n)*(3-A007814(n)/(A007814(n)+2)) - Velin Yanev, Jun 30 2017

Proof: Let m = A007814(m) and decompose n into n = k*2^m. We know from Chai Wah Wu's program below that a(n) = Product(p_i^(e_i-1)*((p_i-1)*e_i+p_i)) where the numbers p_i are the prime factors of n and e_i are the corresponding exponents. Hence a(2n) = 2^m*(m+3)*a(k) = 2^m*(m+3)*a(k). On the other hand, a(n) = 2^(m-1)*(m+2)*a(k). Dividing the first equation by the second yields a(2n)/a(n) = 2*(m+3)/(m+2), which equals 3 - m/(m+2). Hence a(2n) = a(n)*(3 - m/(m+2)). - Ruediger Jehn, Jun 23 2022

Sum_{k=1..n} a(k) ~ 3*n^2/Pi^2 * (log(n) - 1/2 + 2*gamma - 6*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 08 2019

a(n) = Sum_{k=1..n} n/gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021

log(a(n)/n) << log n log log log n/log log n; in particular, a(n) << n^(1+e) for any e > 0. See Broughan link for bounds in terms of omega(n). - Charles R Greathouse IV, Sep 08 2022

a(n) = (1/4)*Sum_{k = 1..4*n} (-1)^k * gcd(k, 4*n) = (1/4) * A344372(2*n). - Peter Bala, Jan 01 2024