A054523 -id:A054523 - cp's OEIS frontend

A129558 A054523 * A129360.

1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 3, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 5, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 4, 0, 1, 0, 0, 0, 0, 0, 1, 3, 3, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 20 2007

Row sums = A062570, phi(2*n): (1, 2, 2, 4, 4, 4, 6, 8, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 0, 1;
  2, 1, 0, 1;
  3, 0, 0, 0, 1;
  1, 1, 1, 0, 0, 1;
  5, 0, 0, 0, 0, 0, 1;
  4, 2, 0, 1, 0, 0, 0, 1;
  4, 0, 1, 0, 0, 0, 0, 0, 1;
  ...

Cf. A054523, A129360, A062570.

A054523 * A129360 as infinite lower triangular matrices.

A143276 Triangle read by rows: A054525 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 5, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...).

Left border = A007431: (1, 0, 1, 1, 3, 0, 5, 2, 4,...).

			First few rows of the triangle =
1;
0, 1;
1, 0, 1;
1, 0, 0, 1;
3, 0, 0, 0, 1;
0, 1, 0, 0, 0, 1;
5, 0, 0, 0, 0, 0, 1;
2, 1, 0, 0, 0, 0, 0, 1;
...

Cf. A000010, A007431, A054523, A054525.

Moebius transform of triangle A054523.

a(17), a(18) corrected by Georg Fischer, May 29 2023

A129555 A054523 * A129372.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Apr 20 2007

Row sums = A002131: (1, 2, 4, 4, 6, 8, 8, 8, 13, 12, ...). Left column = A016741: (1, 1, 3, 2, 5, 3, 7, ...).

			First few rows of the triangle:
  1;
  1, 1;
  3, 0, 1;
  2, 1, 0, 1;
  5, 0, 0, 0, 1;
  3, 3, 1, 0, 0, 1;
  7, 0, 0, 0, 0, 0, 1;
  4, 2, 0, 1, 0, 0, 0, 1;
  9, 0, 3, 0, 0, 0, 0, 0, 1;
  ...

Cf. A054523, A129372, A002131, A026741.

A054523 * A129372 as infinite lower triangular matrices.

A129560 A054523 * A128174.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 2, 0, 1, 7, 0, 1, 0, 1, 5, 4, 1, 1, 0, 1, 13, 0, 1, 0, 1, 0, 1, 9, 6, 1, 2, 0, 1, 0, 1, 19, 0, 3, 0, 1, 0, 1, 0, 1, 13, 10, 1, 2, 1, 1, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, Apr 20 2007

Row sums = A002620: (1, 2, 4, 6, 9, 12, 16, 20, ...). Left column = A106477: (1, 1, 3, 3, 7, 5, 13, 9, 19, ...).

			First few rows of the triangle:
   1;
   1, 1;
   3, 0, 1;
   3, 2, 0, 1;
   7, 0, 1, 0, 1;
   5, 4, 1, 1, 0, 1;
  13, 0, 1, 0, 1, 0, 1;
  ...

Cf. A054523, A128174, A106477, A002620.

A054523 * A128174 as infinite lower triangular matrices.

A129561 A054523 * A115369.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 1, 0, 0, 1, 4, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 20 2007

Row sums = A026741: (1, 1, 3, 3, 2, 5, 3, 7, 4, ...).

			First few rows of the triangle:
  1;
  0, 1;
  2, 0, 1;
  1, 0, 0, 1;
  4, 0, 0, 0, 1;
  0, 2, 0, 0, 0, 1;
  6, 0, 0, 0, 0, 0, 1;
  ...

Cf. A054523, A115369, A026741.

A054523 * A115369 as infinite lower triangular matrices.

A130028 A129686 * A054523.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 2, 0, 1, 6, 0, 1, 0, 1, 4, 3, 1, 1, 0, 1, 10, 0, 0, 0, 1, 0, 1, 6, 4, 1, 1, 0, 1, 0, 1, 12, 0, 2, 0, 0, 0, 1, 0, 1, 8, 6, 0, 1, 1, 0, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, May 02 2007

Row sums = A004277, 1 followed by even terms: (1, 2, 4, 6, 8, 10, ...).

			First few rows of the triangle:
   1;
   1, 1;
   3, 0, 1;
   3, 2, 0, 1;
   6, 0, 1, 0, 1;
   4, 3, 1, 1, 0, 1;
  10, 0, 0, 0, 1, 0, 1;
   6, 4, 1, 1, 0, 1, 0, 1;
  ...

Cf. A129686, A054523, A004277.

A129686 * A054523 as infinite lower triangular matrices, where A129686 = the alternate term operator.

A130313 A000012 * A054523.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 10, 2, 1, 1, 1, 12, 4, 2, 1, 1, 1, 18, 4, 2, 1, 1, 1, 1, 22, 6, 2, 2, 1, 1, 1, 1, 28, 6, 4, 2, 1, 1, 1, 1, 1, 32, 10, 4, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, May 21 2007

Row sums = the triangular series, A000217: (1, 3, 6, 10, ...).

Left column = A002088: (1, 2, 4, 6, 10, 12, 18, ...).

			First few rows of the triangle:
   1;
   2, 1;
   4, 1, 1;
   6, 2, 1, 1;
  10, 2, 1, 1, 1;
  12, 4, 2, 1, 1, 1;
  18, 4, 2, 1, 1, 1, 1;
  ...

Cf. A000012, A054523, A000217.

A000012 * A054523 as infinite lower triangular matrices.

A141294 Triangle read by rows, A054523 * A000012, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 5, 1, 1, 1, 1, 6, 4, 2, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 9, 3, 3, 1, 1, 1, 1, 1, 1, 10, 6, 2, 2, 2, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 8, 6, 4, 2, 2, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 8, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Aug 02 2008

Row sums = A018804: (1, 3, 5, 8, 9, 15, 13, 20,...).

			First few rows of the triangle =
1;
2, 1;
3, 1, 1;
4, 2, 1, 1;
5, 1, 1, 1, 1;
6, 4, 2, 1, 1, 1;
7, 1, 1, 1, 1, 1, 1;
8, 4, 2, 2, 1, 1, 1, 1;
...

Cf. A054523, A018804.

Triangle read by rows, A054523 * A000012, 1<=k<=n. By rows, partial sums starting from the right.

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

Original entry on oeis.org

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

David W. Wilson

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

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

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.

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.

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.

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

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

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

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 *)

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

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

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

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

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

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

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

A102190 Irregular triangle read by rows: coefficients of cycle index polynomial for the cyclic group C_n, Z(C_n,x), multiplied by n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 2, 2, 1, 6, 1, 1, 2, 4, 1, 2, 6, 1, 1, 4, 4, 1, 10, 1, 1, 2, 2, 2, 4, 1, 12, 1, 1, 6, 6, 1, 2, 4, 8, 1, 1, 2, 4, 8, 1, 16, 1, 1, 2, 2, 6, 6, 1, 18, 1, 1, 2, 4, 4, 8, 1, 2, 6, 12, 1, 1, 10, 10, 1, 22, 1, 1, 2, 2, 2, 4, 4, 8, 1, 4, 20, 1, 1, 12, 12, 1, 2, 6, 18, 1, 1, 2, 6, 6

Offset: 1

Wolfdieter Lang, Feb 15 2005

Row n gives the coefficients of x[k]^{n/k} with increasing divisors k of n.
The length of row n is tau(n) = A000005(n) (number of divisors of n, including 1 and n).
See also table A054523 with zeros if k does not divide n, and reversed rows. [Wolfdieter Lang, May 29 2012]

			Array begins:
1: [1],
2: [1, 1],
3: [1, 2],
4: [1, 1, 2],
5: [1, 4],
6: [1, 1, 2, 2],
7: [1, 6], ...
The entry for n=6 is obtained as follows:
Z(C_6,x)=(1*x[1]^6 + 1*x[2]^3 + 2*x[3]^2 + 2*x[6]^1)/6.
a(6,1)=phi(1)=1, a(6,2)=phi(2)=1, a(6,3)=phi(3)=2, a(6,4)=phi(6)=2.

N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144-184 (see Example 5.7).
F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 181 and 184.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).

Wolfdieter Lang, Table of n, a(n) for n = 1..7069 (suggested by T. D. Noe, Nov 16 2015)
W. Lang, More terms and comments
Carl Pomerance, Lola Thompson, Andreas Weingartner, On integers n for which X^n-1 has a divisor of every degree, arXiv:1511.03357 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cycle Index.

Cf. A054523.

k[n_, m_] := Divisors[n][[m]]; a[n_, m_] := EulerPhi[k[n, m]]; Flatten[Table[a[n, m], {n, 1, 28}, {m, 1, DivisorSigma[0, n]}]] (* Jean-François Alcover, Jul 25 2011, after given formula *)
row[n_] := If[n == 1, {1}, n List @@ CycleIndexPolynomial[CyclicGroup[n], Array[x, n]] /. x[] -> 1]; Array[row, 30] // Flatten (* _Jean-François Alcover, Nov 04 2016 *)

tabf(nn) = for (n=1, nn, print(apply(x->eulerphi(x), divisors(n)))); \\ Michel Marcus, Nov 13 2015

tabf(nn) = for (n=1, nn, print(apply(x->poldegree(x), factor(x^n-1)[,1]))) \\ Michel Marcus, Nov 13 2015

a(n, m) = phi(k(m)), m=1..tau(n), n>=1, with k(m) the m-th divisor of n, written in increasing order.

Z(C_n, x):=sum(sum(phi(k)*x[k]^{n/k}, k|n))/n, where phi(n)= A000010(n) (Euler's totient function) and k|n means 'k divides n'. Cf. Harary-Palmer reference and MathWorld link.

cp's OEIS Frontend

A129558 A054523 * A129360.

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A143276 Triangle read by rows: A054525 * A054523 as infinite lower triangular matrices.

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A129555 A054523 * A129372.

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A129560 A054523 * A128174.

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A129561 A054523 * A115369.

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A130028 A129686 * A054523.

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A130313 A000012 * A054523.

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A141294 Triangle read by rows, A054523 * A000012, 1<=k<=n.

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

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