A143104 -id:A143104 - cp's OEIS frontend

A008683 Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1

Offset: 1

N. J. A. Sloane

Moebius inversion: f(n) = Sum_{d|n} g(d) for all n <=> g(n) = Sum_{d|n} mu(d)*f(n/d) for all n.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3, 1).
A008683 = A140579^(-1) * A140664. - Gary W. Adamson, May 20 2008
Coons & Borwein prove that Sum_{n>=1} mu(n) z^n is transcendental. - Jonathan Vos Post, Jun 11 2008; edited by Charles R Greathouse IV, Sep 06 2017
Equals row sums of triangle A144735 (the square of triangle A054533). - Gary W. Adamson, Sep 20 2008
Conjecture: a(n) is the determinant of Redheffer matrix A143104 where T(n, n) = 0. Verified for the first 50 terms. - Mats Granvik, Jul 25 2008
From Mats Granvik, Dec 06 2008: (Start)
The Editorial Office of the Journal of Number Theory kindly provided (via B. Conrey) the following proof of the conjecture: Let A be A143104 and B be A143104 where T(n, n) = 0.
"Suppose you expand det(B_n) along the bottom row. There is only a 1 in the first position and so the answer is (-1)^n times det(C_{n-1}) say, where C_{n-1} is the (n-1) by (n-1) matrix obtained from B_n by deleting the first column and the last row. Now the determinant of the Redheffer matrix is det(A_n) = M(n) where M(n) is the sum of mu(m) for 1 <= m <= n. Expanding det(A_n) along the bottom row, we see that det(A_n) = (-1)^n * det(C_{n-1}) + M(n-1). So we have det(B_n) = (-1)^n * det(C_{n-1}) = det(A_n) - M(n-1) = M(n) - M(n-1) = mu(n)." (End)
Conjecture: Consider the table A051731 and treat 1 as a divisor. Move the value in the lower right corner vertically to a divisor position in the transpose of the table and you will find that the determinant is the Moebius function. The number of permutation matrices that contribute to the Moebius function appears to be A074206. - Mats Granvik, Dec 08 2008
Convolved with A152902 = A000027, the natural numbers. - Gary W. Adamson, Dec 14 2008
[Pickover, p. 226]: "The probability that a number falls in the -1 mailbox turns out to be 3/Pi^2 - the same probability as for falling in the +1 mailbox". - Gary W. Adamson, Aug 13 2009
Let A = A176890 and B = A * A * ... * A, then the leftmost column in matrix B converges to the Moebius function. - Mats Granvik, Gary W. Adamson, Apr 28 2010 and May 28 2020
Equals row sums of triangle A176918. - Gary W. Adamson, Apr 29 2010
Calculate matrix powers: A175992^0 - A175992^1 + A175992^2 - A175992^3 + A175992^4 - ... Then the Mobius function is found in the first column. Compare this to the binomial series for (1+x)^-1 = 1 - x + x^2 - x^3 + x^4 - ... . - Mats Granvik, Gary W. Adamson, Dec 06 2010
From Richard L. Ollerton, May 08 2021: (Start)
Formulas for the numerous OEIS entries involving the Möbius transform (Dirichlet convolution of a(n) and some sequence h(n)) can be derived using the following (n >= 1):
Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.
Use of gcd(n,k)*lcm(n,k) = n*k provides further variations. (End)
Formulas for products corresponding to the sums above are also available for sequences f(n) > 0: Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))). - Richard L. Ollerton, Nov 08 2021

			G.f. = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 64-65.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
Clifford A. Pickover, "The Math Book, from Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics", Sterling Publishing, 2009, p. 226. - Gary W. Adamson, Aug 13 2009
G. Pólya and G. Szegő, Problems and Theorems in Analysis Volume II. Springer_Verlag 1976.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 98-99.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 826.
All Angles, What is the Moebius function?, video, 2024.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 705-707.
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Olivier Bordellès, Some Explicit Estimates for the Mobius Function , J. Int. Seq. 18 (2015) 15.11.1
G. J. Chaitin, Thoughts on the Riemann hypothesis arXiv:math/0306042 [math.HO], 2003.
Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, arXiv:0806.1563 [math.NT], 2008.
Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295.
Tom Edgar, Posets and Möbius Inversion, slides, (2008).
Mats Granvik, Inverse of a triangular matrix using determinants, Inverse of a triangular matrix using matrix multiplication, Inverse of a triangular matrix as a binomial series, The ordinary generating function for the Mobius function.
Keith Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n).
A. F. Möbius, Über eine besondere Art von Umkehrung der Reihen. Journal für die reine und angewandte Mathematik 9 (1832), 105-123.
Ed Pegg Jr., The Mobius function (and squarefree numbers).
Anders Björner and Richard P. Stanley, A combinatorial miscellany.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects Of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238.
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, Jun 05 2014, pp. 105-124.
Gerard Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens.
Eric Weisstein's World of Mathematics, Moebius Function.
Eric Weisstein's World of Mathematics, Redheffer Matrix.
Wikipedia, Moebius function.
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Variants of a(n) are A178536, A181434, A181435.
Cf. A000010, A001221, A008966, A007423, A080847, A002321 (partial sums), A069158, A055615, A129360, A140579, A140664, A140254, A143104, A152902, A206706, A063524, A007427, A007428, A124010, A073776, A074206, A132971, A156552.
Cf. A059956 (Dgf at s=2), A088453 (Dgf at s=3), A215267 (Dgf at s=4), A343308 (Dgf at s=5).

Axiom
```
[moebiusMu(n) for n in 1..100]
```

import Math.NumberTheory.Primes.Factorisation (factorise)
a008683 = mu . snd . unzip . factorise where
mu [] = 1; mu (1:es) = - mu es; mu (_:es) = 0
-- Reinhard Zumkeller, Dec 13 2015, Oct 09 2013

a008683 1 = 1
a008683 n = - sum [a008683 d | d <- [1..(n-1)], n `mod` d == 0]
-- Harry Richman, Jun 13 2025

Magma
```
[ MoebiusMu(n) : n in [1..100]];
```

with(numtheory): A008683 := n->mobius(n);
with(numtheory): [ seq(mobius(n), n=1..100) ];
# Note that older versions of Maple define mobius(0) to be -1.
# This is unwise! Moebius(0) is better left undefined.
with(numtheory):
mu:= proc(n::posint) option remember; `if`(n=1, 1,
       -add(mu(d), d=divisors(n) minus {n}))
     end:
seq(mu(n), n=1..100);  # Alois P. Heinz, Aug 13 2008

Array[ MoebiusMu, 100]
(* Second program: *)
m = 100; A[_] = 0;
Do[A[x_] = x - Sum[A[x^k], {k, 2, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Oct 20 2019, after Ilya Gutkovskiy *)

A008683(n):=moebius(n)$ makelist(A008683(n),n,1,30); /* Martin Ettl, Oct 24 2012 */

PARI
```
a=n->if(n<1,0,moebius(n));
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 - X)[n])};

list(n)=my(v=vector(n,i,1)); forprime(p=2, sqrtint(n), forstep(i=p, n, p, v[i]*=-1); forstep(i=p^2, n, p^2, v[i]=0)); forprime(p=sqrtint(n)+1, n, forstep(i=p, n, p, v[i]*=-1)); v \\ Charles R Greathouse IV, Apr 27 2012

from sympy import mobius
print([mobius(i) for i in range(1, 101)])  # Indranil Ghosh, Mar 18 2017

@cached_function
def mu(n):
    if n < 2: return n
    return -sum(mu(d) for d in divisors(n)[:-1])
# Changing the sign of the sum gives the number of ordered factorizations of n A074206.
print([mu(n) for n in (1..96)])  # Peter Luschny, Dec 26 2016

Sum_{d|n} mu(d) = 1 if n = 1 else 0.
Dirichlet generating function: Sum_{n >= 1} mu(n)/n^s = 1/zeta(s). Also Sum_{n >= 1} mu(n)*x^n/(1-x^n) = x.
In particular, Sum_{n > 0} mu(n)/n = 0. - Franklin T. Adams-Watters, Jun 20 2014
phi(n) = Sum_{d|n} mu(d)*n/d.
a(n) = A091219(A091202(n)).
Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1. - David W. Wilson, Aug 01 2001
abs(a(n)) = Sum_{d|n} 2^A001221(d)*a(n/d). - Benoit Cloitre, Apr 05 2002
Sum_{d|n} (-1)^(n/d)*mobius(d) = 0 for n > 2. - Emeric Deutsch, Jan 28 2005
a(n) = (-1)^omega(n) * 0^(bigomega(n) - omega(n)) for n > 0, where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Dirichlet generating function for the absolute value: zeta(s)/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
mu(n) = A129360(n) * (1, -1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007
mu(n) = -Sum_{d < n, d|n} mu(d) if n > 1 and mu(1) = 1. - Alois P. Heinz, Aug 13 2008
a(n) = A174725(n) - A174726(n). - Mats Granvik, Mar 28 2010
a(n) = first column in the matrix inverse of a triangular table with the definition: T(1, 1) = 1, n > 1: T(n, 1) is any number or sequence, k = 2: T(n, 2) = T(n, k-1) - T(n-1, k), k > 2 and n >= k: T(n,k) = (Sum_{i = 1..k-1} T(n-i, k-1)) - (Sum_{i = 1..k-1} T(n-i, k)). - Mats Granvik, Jun 12 2010
Product_{n >= 1} (1-x^n)^(-a(n)/n) = exp(x) (product form of the exponential function). - Joerg Arndt, May 13 2011
a(n) = Sum_{k=1..n, gcd(k,n)=1} exp(2*Pi*i*k/n), the sum over the primitive n-th roots of unity. See the Apostol reference, p. 48, Exercise 14 (b). - Wolfdieter Lang, Jun 13 2011
mu(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n. (conjecture). - Mats Granvik, Nov 20 2011
Sum_{k=1..n} a(k)*floor(n/k) = 1 for n >= 1. - Peter Luschny, Feb 10 2012
a(n) = floor(omega(n)/bigomega(n))*(-1)^omega(n) = floor(A001221(n)/A001222(n))*(-1)^A001221(n). - Enrique Pérez Herrero, Apr 27 2012
Multiplicative with a(p^e) = binomial(1, e) * (-1)^e. - Enrique Pérez Herrero, Jan 19 2013
G.f. A(x) satisfies: x^2/A(x) = Sum_{n>=1} A( x^(2*n)/A(x)^n ). - Paul D. Hanna, Apr 19 2016
a(n) = -A008966(n)*A008836(n)/(-1)^A005361(n) = -floor(rad(n)/n)Lambda(n)/(-1)^tau(n/rad(n)). - Anthony Browne, May 17 2016
a(n) = Kronecker delta of A001221(n) and A001222(n) (which is A008966) multiplied by A008836(n). - Eric Desbiaux, Mar 15 2017
a(n) = A132971(A156552(n)). - Antti Karttunen, May 30 2017
Conjecture: a(n) = Sum_{k>=0} (-1)^(k-1)*binomial(A001222(n)-1, k)*binomial(A001221(n)-1+k, k), for n > 1. Verified for the first 100000 terms. - Mats Granvik, Sep 08 2018
From Peter Bala, Mar 15 2019: (Start)
Sum_{n >= 1} mu(n)*x^n/(1 + x^n) = x - 2*x^2. See, for example, Pólya and Szegő, Part V111, Chap. 1, No. 71.
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 - x^n) = x + 2*(x^2 + x^4 + x^8 + x^16 + ...).
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 + x^n) = x - 2*(x^4 + x^8 + x^16 + x^32 + ...).
Sum_{n >= 1} |mu(n)|*x^n/(1 - x^n) = Sum_{n >= 1} (2^w(n))*x^n, where w(n) is the number of different prime factors of n (Hardy and Wright, Chapter XVI, Theorem 264).
Sum_{n odd} |mu(n)|*x^n/(1 + x^(2*n)) = Sum_{n in S_1} (2^w_1(n))*x^n, where S_1 = {1, 5, 13, 17, 25, 29, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 1 (mod 4), and w_1(n) is the number of different prime factors p = 1 (mod 4) of n.
Sum_{n odd} (-1)^((n-1)/2)*mu(n)*x^n/(1 - x^(2*n)) = Sum_{n in S_3} (2^w_3(n))*x^n, where S_3 = {1, 3, 7, 9, 11, 19, 21, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 3 (mod 4), and where w_3(n) is the number of different prime factors p = 3 (mod 4) of n. (End)
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, May 11 2019
a(n) = sign(A023900(n)) * [A007947(n) = n] where [] is the Iverson bracket. - I. V. Serov, May 15 2019
a(n) = Sum_{k = 1..n} gcd(k, n)*a(gcd(k, n)) = Sum_{d divides n} a(d)*d*phi(n/d). - Peter Bala, Jan 16 2024

A083058 Number of eigenvalues equal to 1 of n X n matrix A(i,j)=1 if j=1 or i divides j.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset: 1

Michael Somos, Apr 18 2003

nonn

All numbers occur at least once, but terms > 1 of A000295 appear twice. - Robert G. Wilson v, Apr 19 2006
It appears that a(n) = Sum_{k=0..n-1} (1 + (-1)^A000108(k))/2 (n > 1). - Paul Barry, Mar 31 2008
Barry's observation above is true because A000108 obtains odd values only at points (2^j)-1 (A000225) and here the repeated values (A000295) occur precisely at positions given by A000225 and A000079. - Antti Karttunen, Aug 17 2013
a(n)+1 gives a lower bound for nonzero terms of A228086 and A228087. - Antti Karttunen, Aug 17 2013

Antti Karttunen, Table of n, a(n) for n = 1..8192
J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347.
Will Dana, Eigenvalues of the Redheffer Matrix and their relation to the Mertens function (2015), Theorem 5.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions.
Ralf Stephan, Table of generating functions.

Cf. A002321, A070939, A143104.

A083058 := proc(n)
    if n = 1 then
        1;
    else
        n-floor(log[2](n))-1 ;
    end if;
end proc:
seq(A083058(n),n=1..40) ; # R. J. Mathar, Jul 23 2017

a[1] = 1; a[n_] := n - Floor[Log[2, n]] - 1;
Array[a, 100] (* Jean-François Alcover, Feb 27 2019 *)

a(n)=if(n<2,n>0,n-floor(log(n)/log(2))-1)

a(n)= if(n<1, 0, valuation( subst( charpoly( matrix(n, n, i, j, (j==1) || (0==j%i))), x, x+1), x))

def a(n): return n - n.bit_length() + (n == 1)  # Matthew Andres Moreno, Jan 04 2024

(define (A083058 n) (if (< n 2) n (- n (A070939 n)))) ;; Antti Karttunen, Aug 17 2013

a(n) = n - A070939(n), n > 1.
a(1)=1, else a(n)=b(n) with b(0)=0, b(2n)=b(n)+n-1, b(2n+1)=b(n)+n. - Ralf Stephan, Oct 11 2003
Except for a(1), a(n) = n - 1 - floor(log(2,n)). - Robert G. Wilson v, Apr 19 2006
It seems that a(n) = A182220(n+1)-1 for all n > 1. - Antti Karttunen, Aug 17 2013

A161886 Number of nonzero elements in the n X n Redheffer matrix.

Original entry on oeis.org

1, 4, 7, 11, 14, 19, 22, 27, 31, 36, 39, 46, 49, 54, 59, 65, 68, 75, 78, 85, 90, 95, 98, 107, 111, 116, 121, 128, 131, 140, 143, 150, 155, 160, 165, 175, 178, 183, 188, 197, 200, 209, 212, 219, 226, 231, 234, 245, 249, 256, 261, 268, 271, 280, 285, 294, 299, 304

Offset: 1

Mats Granvik, Jun 21 2009

nonn

			The 4x4 Redheffer matrix:
  1,1,1,1
  1,1,0,0
  1,0,1,0
  1,1,0,1
contains 11 nonzero elements.

Cf. A143104, A006590, A000005.

A161886[n_] := Plus @@ Table[DivisorSigma[0, i], {i, 1, n}] + n - 1 (* Enrique Pérez Herrero, Sep 25 2009 *)
A161886[n_] := Total[Table[ Boole[Divisible[i, j] || (i == 1)], {i, 1, n}, {j, 1, n}], Infinity] (* Enrique Pérez Herrero, Sep 25 2009 *)
A161889[n_] := Plus @@ Plus @@ Table[Boole[Divisible[i, j] || (i == 1)], {i, 1, n}, {j, 1, n}] (* Enrique Pérez Herrero, Sep 28 2009 *)
A161889[n_] := Sum[Ceiling[n/i], {i, 1, n}] + DivisorSigma[0, n] - 1 (* Enrique Pérez Herrero, Sep 28 2009 *)

from math import isqrt
def A161886(n): return (lambda m: 2*sum(n//k for k in range(1, m+1))+n-1-m*m)(isqrt(n)) # Chai Wah Wu, Oct 09 2021

a(n) = A006590(n)+A000005(n)-1. [Enrique Pérez Herrero, Sep 28 2009]
a(n) = A006218(n)+n-1. [Enrique Pérez Herrero, Sep 25 2009]
a(1) = 1, a(n) = a(n-1) + A000005(n) + 1 for n > 1. a(1) = 1, a(n) = A006218(n+1) - A000005(n+1) + n - 1 = A006218(n+1) + A049820(n+1) - 2 = A006590(n+1) - 2 for n > 1. [Jaroslav Krizek, Nov 08 2009]

Edited by N. J. A. Sloane, Jun 26 2009

A077478 Rectangular array R read by antidiagonals: R(i,j) is the number of integers k that divide both i and j (i >= 1, j >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1

Offset: 1

Clark Kimberling, Nov 08 2002

Antidiagonal sums of R, alias row sums of T, are essentially A065608. Diagonal elements of R comprise A000203 (sums of divisors of n).
Antidiagonals of an array formed by A051731 * A051731 (transposed). - Gary W. Adamson, Nov 12 2007
If R(n) is the n X n Redheffer matrix (A143104) and Rt(n) is its transposed matrix, then this sequence seems to be formed by R(n)*Rt(n). - Enrique Pérez Herrero, Feb 21 2012

			First few rows of the array R are:
  1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 1, ...
  1, 1, 2, 1, 1, 2, 1, ...
  1, 2, 1, 3, 1, 2, 1, ...
  1, 1, 1, 1, 2, 1, 1, ...
  1, 2, 2, 2, 1, 4, 1, ...
  ...
First few rows of the triangle T are:
  1;
  1, 1;
  1, 2, 1;
  1, 1, 1, 1;
  1, 2, 2, 2, 1;
  1, 1, 1, 1, 1, 1;
  1, 2, 1, 3, 1, 3, 1;
  1, 1, 2, 1, 1, 2, 1, 1;
  1, 2, 1, 2, 2, 2, 1, 2, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1;
  ...
R(4,2)=2 since 1|2, 1|4 and 2|2, 2|4.

Stefano Spezia, First 150 antidiagonals of the array, flattened

Cf. A051194, A077049, A077051.

Cf. A051731, A065608.

T[n_,k_]:=DivisorSigma[0,GCD[n,k]]; Flatten[Table[T[n-k+1,k],{n,14},{k,n}]] (* Stefano Spezia, May 23 2021 *)

R=U*V, where U and V are the summatory matrices (A077049, A077051). The triangle T(n, k) formed by antidiagonals: T(n, k)=tau(gcd(k, n+1-k)) for 1<=k<=n, where tau(m)=A000005(m). [Corrected by Leroy Quet, Apr 08 2009]

Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} tau(gcd(n,k))/n^s/k^c = zeta(s)*zeta(c)* zeta(s + c). - Mats Granvik, May 19 2021

Edited by N. J. A. Sloane, Jan 11 2009

A143142 Square array T(n,k) read by antidiagonals up. Mobius function interleaved with k-1 zeros in each column. Redheffer type matrix.

Original entry on oeis.org

1, -1, 1, -1, 1, 1, 0, 0, 0, 1, -1, -1, 1, 0, 1, 1, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Mats Granvik, Gary W. Adamson and Roger L. Bagula, Jul 27 2008, Sep 21 2008

Determinant of the n times n matrix with T(n,n)=0 is equal to 1 for every such matrix. Tested up to n=13.

For elements where row index is greater than column index, every negative number appears to give a positive contribution while every positive number appears to give a negative contribution to the determinant of the matrix. The size of a such contribution appears to be 1 when replacing the number with a zero. [From Mats Granvik, Sep 22 2008]

			Determinant of:
1
is equal to 1.
Determinant of:
1,1
-1,1
is equal to 2.
Determinant of:
1,1,1
-1,1,0
-1,0,1
is equal to 3.
Determinant of:
1,1,1,1
-1,1,0,0
-1,0,1,0
0,-1,0,1
is equal to 4.
Determinant of:
1,1,1,1,1
-1,1,0,0,0
-1,0,1,0,0
0,-1,0,1,0
-1,0,0,0,1
is equal to 5.
Determinant of:
1,1,1,1,1,1
-1,1,0,0,0,0
-1,0,1,0,0,0
0,-1,0,1,0,0
-1,0,0,0,1,0
1,-1,-1,0,0,1
is equal to 6.
Determinant of:
1,1,1,1,1,1,1
-1,1,0,0,0,0,0
-1,0,1,0,0,0,0
0,-1,0,1,0,0,0
-1,0,0,0,1,0,0
1,-1,-1,0,0,1,0
-1,0,0,0,0,0,1
is equal to 7.

Solve My Math, Matrix Calculator.
E. W. Weisstein, Redheffer Matrix.
Wikipedia, Moebius Function Matrix.

Cf. A008683, A143104, A000027.

Cf. A054525. [From Mats Granvik, Sep 21 2008]

=if(mod(row();column())=0;lookup(row()/column();A000027;A008683);if(row()=1;1;0))

A144193 Square array (5 X 5) read by rows.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 5th value of the moebius function. Determinant of matrix is equal to -1. The array is symmetrical.

			The array is:
0,0,0,0,1
0,1,0,0,0
0,0,1,0,0
0,0,0,1,0
1,0,0,0,0

J. Davies, Permutation Calculator
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033.

A144195 Square array (6 X 6) read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 6th value of the Möbius function. Determinant of matrix is equal to 1. The array is asymmetrical.

			The array is:
0,0,0,0,0,1
1,0,0,0,0,0
0,0,1,0,0,0
0,0,0,1,0,0
0,0,0,0,1,0
0,1,0,0,0,0

J. Davies, Permutation Calculator
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033, A144194, A144196.

A144196 Square array (6 X 6) read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 6th value of the Möbius function. Determinant of matrix is equal to 1. The array is asymmetrical.

			The array is:
0,0,0,0,0,1
0,1,0,0,0,0
1,0,0,0,0,0
0,0,0,1,0,0
0,0,0,0,1,0
0,0,1,0,0,0

J. Davies, Permutation Calculator
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033, A144195, A144197.

A144197 Square array 7 x 7 read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 7th value of the Möbius function. Determinant of matrix is equal to -1. The array is symmetrical.

			The array is:
0,0,0,0,0,0,1
0,1,0,0,0,0,0
0,0,1,0,0,0,0
0,0,0,1,0,0,0
0,0,0,0,1,0,0
0,0,0,0,0,1,0
1,0,0,0,0,0,0

Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033, A144196, A144198.

A144198 Square array (8 X 8) read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 8th value of the Möbius function. Determinant of matrix is equal to -1. The array is symmetrical.

			The array is:
0,0,0,0,0,0,0,1
0,1,0,0,0,0,0,0
0,0,1,0,0,0,0,0
0,0,0,1,0,0,0,0
0,0,0,0,1,0,0,0
0,0,0,0,0,1,0,0
0,0,0,0,0,0,1,0
1,0,0,0,0,0,0,0

H. S. Wilf The Redheffer matrix of a partially ordered set

Cf. A143104, A002033, A144197, A144199.

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A143142 Square array T(n,k) read by antidiagonals up. Mobius function interleaved with k-1 zeros in each column. Redheffer type matrix.

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A144193 Square array (5 X 5) read by rows.

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