A054525 -id:A054525 - cp's OEIS frontend

A168532 Triangle read by rows, A054525 * A168021.

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 6, 0, 0, 0, 1, 7, 2, 1, 0, 0, 1, 14, 0, 0, 0, 0, 0, 1, 17, 3, 0, 1, 0, 0, 0, 1, 27, 0, 2, 0, 0, 0, 0, 0, 1, 34, 6, 0, 0, 1, 0, 0, 0, 0, 1, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 63, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Nov 28 2009

Row sums = A000041 starting (1, 2, 3, 5, 7, 11, 15, ...).

T(n,k) is the number of partitions of n into parts with GCD = k. - Alois P. Heinz, Jun 06 2013

			First few rows of the triangle:
    1;
    1,  1;
    2,  0, 1;
    3,  1, 0, 1;
    6,  0, 0, 0, 1;
    7,  2, 1, 0, 0, 1;
   14,  0, 0, 0, 0, 0, 1;
   17,  3, 0, 1, 0, 0, 0, 1;
   27,  0, 2, 0, 0, 0, 0, 0, 1;
   34,  6, 0, 0, 1, 0, 0, 0, 0, 1;
   55,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   63,  7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
  100,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  119, 14, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
  167,  0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  209, 17, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  296,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A168021, A000837.

Cf. A256067 (the same for LCM).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^igcd(t, i),
      t=0..degree(p)))(add(b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..17);  # Alois P. Heinz, Mar 29 2015

b[n_, i_] := b[n, i] = If[n==0, 1, If[i==1, x, b[n, i-1] + Function[{p}, Sum[Coefficient[p, x, t]*x^GCD[t, i], {t, 0, Exponent[p, x]}]][Sum[b[n - i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 17}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

Mobius transform of triangle A168021 = an infinite lower triangular matrix with aerated variants of A000837 in each column; where A000837 = the Mobius transform of the partition numbers, A000041.

Corrected and extended by Alois P. Heinz, Jun 06 2013

A129360 A054525 * A115361.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Apr 10 2007

Row sums = A209229 (1, 1, 0, 1, 0, 0, 0, 1, ...).

A129353 = the inverse Möbius transform of A115361.

			First few rows of the triangle are:
   1;
   0,  1;
  -1,  0,  1;
   0,  0,  0,  1;
  -1,  0,  0,  0,  1;
   0, -1,  0,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A087003 (Moebius transform of A209229).
Row sums are A209229.
Cf. A054525, A115361, A129353, A001511, A103994, A129501.

tabl(nn) = {Tm = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); Tr = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = Tr^(-1); Tp = Tm*Ti; for (n=1, nn, for (k=1, n, print1(Tp[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

T(n, k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

Moebius transform of A115361.

T(n,k) = A087003(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

More terms from Michel Marcus, Mar 28 2015

Offset changed by Andrew Howroyd, Aug 03 2018

A143519 Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)A010051(d); A054525 A010051.

Original entry on oeis.org

0, 1, 1, -1, 1, -2, 1, 0, -1, -2, 1, 1, 1, -2, -2, 0, 1, 1, 1, 1, -2, -2, 1, 0, -1, -2, 0, 1, 1, 3, 1, 0, -2, -2, -2, 0, 1, -2, -2, 0, 1, 3, 1, 1, 1, -2, 1, 0, -1, 1, -2, 1, 1, 0, -2, 0, -2, -2, 1, -1, 1, -2, 1, 0, -2, 3, 1, 1, -2, 3, 1, 0, 1, -2, 1, 1, -2, 3, 1, 0, 0, -2, 1, -1, -2, -2, -2, 0, 1

Offset: 1

Gary W. Adamson, Aug 22 2008

sign

A010051 = A051731 * A143519 (since A051731 = the inverse Mobius transform).

A000720(n) = Sum_{k=1..n} a(k) floor(n/k) where A000720(n) is the number of primes <= n. - Steven Foster Clark, May 25 2018

			a(4) = -1 since row 4 of triangle A043518 = (0, -1, 0, 0).
a(4) = -1 = (0, -1, 0, 1) dot (0, 1, 1, 0), where (0, -1, 0, 1) = row 4 of A054525 and A010051 = (0, 1, 1, 0, 1, 0, 1, 0, ...).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A008683, A010051, A143518, A054525, A137851.

Table[Sum[MoebiusMu[n/d] Boole[PrimeQ@ d], {d, Divisors@ n}], {n, 89}] (* Michael De Vlieger, Jul 19 2017 *)

A143519(n) = sumdiv(n,d,isprime(d)*moebius(n/d)); \\ (After Luschny's Sage-code) - Antti Karttunen, Jul 19 2017

def A143519(n) :
    D = filter(is_prime, divisors(n))
    return add(moebius(n/d) for d in D)
[A143519(n) for n in (1..89)]   # Peter Luschny, Feb 01 2012

Mobius transform of A010051, the characteristic function of the primes.
Row sums of triangle A143518.
a(n) = Sum_{d|n} A010051(d)*A008683(n/d). - Antti Karttunen, Jul 19 2017
a(n) = Sum_{a*b*c=n} omega(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

More terms from R. J. Mathar, Jan 19 2009

A134541 Triangle read by rows: A000012 * A054525 regarded as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, -1, 0, 1, 1, -2, 0, 1, 1, 1, -1, -1, 0, 1, 1, 1, -2, -1, 0, 1, 1, 1, 1, -2, -1, 0, 0, 1, 1, 1, 1, -2, -1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -1, 0, 0, 1, 1, 1, 1, 1, -2, -2, -1, 0, 0, 1, 1, 1, 1, 1, 1, -2, -1, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = 1.
Left border = A002321, the Mertens function.
A134541 * [1,2,3,...] = A002088: (1, 2, 4, 6, 10, 12, 18, 22, ...).

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

Cf. A000012, A054525, A002321, A002088, A134541.

Matrix inverse of A176702. - Mats Granvik, Apr 24 2010

Clear[t, s, n, k, z, x]; z = 1; nn = 10; t[n_, k_] := t[n, k] = If[n >= k, If[k == 1, 1 - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], t[Floor[n/k], 1]], 0]; Flatten[Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Jul 22 2012 *) (* updated Mats Granvik, Apr 10 2016 *)

Recurrence: T(n, k) = If n >= k then If k = 1 then 1 - Sum_{i=1..n-1} T(n, k + i)/(i + 1)^(s - 1) else T(floor(n/k) else 1)) else 0). - Mats Granvik, Apr 17 2016

More terms from Amiram Eldar, Jun 09 2024

A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 0, 6, 4, 4, 0, 8, 5, 0, 0, 0, 20, 6, 6, 12, 0, 0, 12, 7, 0, 0, 0, 0, 0, 42, 8, 8, 0, 16, 0, 0, 0, 32, 9, 0, 18, 0, 0, 0, 0, 0, 54, 10, 10, 0, 0, 40, 0, 0, 0, 0, 40

Offset: 1

Gary W. Adamson, Jan 15 2007

Mobius transform of A127481.

			First few rows of the triangle are:
1;
2, 2;
3, 0, 6;
4, 4, 0, 8;
5, 0, 0, 0, 20;
6, 6, 12, 0, 0, 12;
7, 0, 0, 0, 0, 0, 42;
8, 8, 0, 16, 0, 0, 0, 32;
...

Cf. A054522, A127093, A062952, A002618.

Cf. A127093, A127481, A054525.

A127466 := proc(n,k)
    add( A054525(n,j)*A127481(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 06 2013

Sum_{k=1..n} T(n,k) = n^2.

T(n,n) = A002618(n) = n*phi(n).

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A137851 a(n) = A054525(n) * A061397(n).

Original entry on oeis.org

0, 2, 3, -2, 5, -5, 7, 0, -3, -7, 11, 2, 13, -9, -8, 0, 17, 3, 19, 2, -10, -13, 23, 0, -5, -15, 0, 2, 29, 10, 31, 0, -14, -19, -12, 0, 37, -21, -16, 0, 41, 12, 43, 2, 3, -25, 47, 0, -7, 5, -20, 2, 53, 0, -16, 0, -22, -31, 59, -2, 61, -33, 3, 0, -18, 16, 67, 2, -26, 14, 71, 0, 73, -39, 5, 2, -18, 18, 79, 0, 0, -43, 83, -2, -22, -45, -32, 0

Offset: 1

Gary W. Adamson, Feb 14 2008

Equals row sums of triangle A143517. - Gary W. Adamson, Aug 22 2008

			a(4) = -2 = (0, -1, 0, 1) dot (0, 2, 3, 0), where (0, -1, 0, 1) = row 4 of the Möbius triangle A054525 and (0, 2, 3, 0) = the first 4 terms of A061397.

Cf. A061397, A054525, A143517, A143519.

A061397 := proc(n) if isprime(n) then n; else 0 ; fi ; end: A054525 := proc(n,k) if n mod k = 0 then numtheory[mobius](n/k); else 0; fi ; end: A137851 := proc(n) local k ; add(A061397(k)* A054525(n,k),k=1..n) ; end: seq(A137851(n),n=1..120) ; # R. J. Mathar, May 23 2008

a[n_] := If[n == 1, 0, With[{p = FactorInteger[n][[All, 1]]}, p*MoebiusMu[n/p] // Total]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 13 2023 *)

def A137851(n):
    return add(d*moebius(n//d) for d in divisors(n) if is_prime(d))
[A137851(n) for n in (1..88)] # Peter Luschny, Feb 01 2012

A054525 * A061397 = Möbius transform of [0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, ...].
Dirichlet g.f.: primezeta(s-1)/zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n} p*mu(n/p), where p is prime. - Ridouane Oudra, Nov 12 2019

More terms from R. J. Mathar, May 23 2008

A156833 A054525 * A156348 * [1,2,3,...].

Original entry on oeis.org

1, 2, 3, 6, 5, 16, 7, 24, 24, 38, 11, 103, 13, 68, 127, 144, 17, 261, 19, 404, 291, 152, 23, 994, 370, 206, 540, 1093, 29, 2195, 31, 1584, 943, 338, 2543, 4808, 37, 416, 1479, 7371, 41, 7929, 43, 4691, 8976, 596, 47, 18876, 6510, 11035, 3091

Offset: 1

Gary W. Adamson, Feb 16 2009

nonn

Conjecture: for n>1, a(n) = n iff n is prime.

Companion to A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19,...).

			a(4) = 6 since first 4 terms of A156348 * [1, 2, 3, 4,...] = (1, 3, 4, 9);
Then (1, 3, 4, 9) dot (0, -1, 0, 1) = (0 - 3 + 0 + 9) = 6. Row 4 of A054525 = (0, -1, 0, 1).

Cf. A156348, A054525, A156834

A156833T := proc(n,k)
    add(A054525(n,j)*A156348(j,k),j=k..n) ;
end proc:
A156833 := proc(n)
    add(A156833T(n,k)*k,k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013

A054525[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0];
A156348[n_, k_] := Which[k < 1 || k > n, 0, Mod[n, k] == 0, Binomial[n/k - 2 + k, k - 1], True, 0];
T[n_, k_] := Sum[A054525[n, j]*A156348[j, k], {j, k, n}];
a[n_] := Sum[T[n, k]*k, {k, 1, n}];
Table[a[n], {n, 1, 51}] (* Jean-François Alcover, Oct 15 2023 *)

A054525 * A156348 * [1,2,3,...]

Extended beyond a(14) by R. J. Mathar, Mar 03 2013

A125093 Triangle T(n,k) = n*A054525(n,k) read by rows.

Original entry on oeis.org

1, -2, 2, -3, 0, 3, 0, -4, 0, 4, -5, 0, 0, 0, 5, 6, -6, -6, 0, 0, 6, -7, 0, 0, 0, 0, 0, 7, 0, 0, 0, -8, 0, 0, 0, 8, 0, 0, -9, 0, 0, 0, 0, 0, 9, 10, -10, 0, 0, -10, 0, 0, 0, 0, 10, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 12, 0, -12, 0, -12, 0, 0, 0, 0, 0, 12, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, -14, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 14, 15, 0, -15, 0

Offset: 1

Gary W. Adamson, Jan 22 2007

			First few rows of the triangle are:
1;
-2, 2;
-3, 0, 3;
0, -4, 0, 4;
-5, 0, 0, 0, 5;
6, -6, -6, 0, 0, 6;
-7, 0, 0, 0, 0, 0, 7;
...

Cf. A055615, A051731, A127648.

A125093 := proc(n,k) if n = k then n; elif n mod k = 0 then n*numtheory[mobius](n/k) ; else 0; end if; end proc:
seq(seq(A125093(n,k),k=1..n),n=1..16) ; # R. J. Mathar, Apr 10 2011

T(n,1) = n*mu(n) = A055615(n).

Offset and definition corrected by R. J. Mathar, Apr 10 2011

A127448 Triangle T(n,k) read by rows: matrix product A054525 * A127648.

Original entry on oeis.org

1, -1, 2, -1, 0, 3, 0, -2, 0, 4, -1, 0, 0, 0, 5, 1, -2, -3, 0, 0, 6, -1, 0, 0, 0, 0, 0, 7, 0, 0, 0, -4, 0, 0, 0, 8, 0, 0, -3, 0, 0, 0, 0, 0, 9, 1, -2, 0, 0, -5, 0, 0, 0, 0, 10, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 2, 0, -4, 0, -6, 0, 0, 0, 0, 0, 12, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 14 2007

			First few rows of the triangle are;
1;
-1, 2;
-1, 0, 3;
0, -2, 0, 4;
-1, 0, 0, 0, 5;
1, -2, -3, 0, 0, 6;
-1, 0, 0, 0, 0, 0, 7;
0, 0, 0, -4, 0, 0, 0, 8;
0, 0, -3, 0, 0, 0, 0, 0, 9;
1, -2, 0, 0,-5, 0, 0, 0, 0, 10;
...

Cf. A000010, A008683, A051731.

A127648 := proc(n,k) if n = k then n; else 0 ; fi; end:
A054525 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi; end:
A127448 := proc(n,k) add( A054525(n,j)*A127648(j,k), j=k..n) ; end: seq(seq( A127448(n,k),k=1..n),n=1..15) ;

T(n,k) = sum _{j=k..n} A054525(n,j)*A127648(j,k) = k*A054525(n,k).
sum_{k=1..n} T(n,k) = A000010(n) (row sums).
T(n,1) = A008683(n).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009
Corrected A-number typo in a formula - R. J. Mathar, Sep 17 2009
Corrected last example line by John Mason, Jan 07 2015

A127638 A054525 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

Original entry on oeis.org

2, -2, 3, -2, 0, 5, 0, -3, 0, 7, -2, 0, 0, 0, 11, 2, -3, -5, 0, 0, 13, -2, 0, 0, 0, 0, 0, 17, 0, 0, 0, -7, 0, 0, 0, 19, 0, 0, -5, 0, 0, 0, 0, 0, 23, 2, -3, 0, 0, -11, 0, 0, 0, 0, 29, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 0, 3, 0, -7, 0, -13, 0, 0, 0, 0, 0, 37, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 2, -3, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 43, 2, 0, -5, 0, -11, 0, 0

Offset: 1

Gary W. Adamson, Jan 21 2007

Right diagonal = primes: (2, 3, 5, 7, ...). Row sums = the Mobius transform of primes, A007444: (2, 1, 3, 4, 9, 7, ...).

			First few rows of the triangle:
   2;
  -2,  3;
  -2,  0,  5;
   0, -3,  0, 7;
  -2,  0,  0, 0, 11;
   2, -3, -5, 0,  0, 13;
  ...

Cf. A007444, A054525, A127639.

A054525 := proc(n,k) if n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi ; end: A127648 := proc(n,k) A054525(n,k)*ithprime(k) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d,", A127648(n,k)) ; od ; od ; # R. J. Mathar, Mar 14 2007

More terms from R. J. Mathar, Mar 14 2007

cp's OEIS Frontend

A168532 Triangle read by rows, A054525 * A168021.

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A129360 A054525 * A115361.

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A143519 Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)*A010051(d); A054525 * A010051.

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A134541 Triangle read by rows: A000012 * A054525 regarded as infinite lower triangular matrices.

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A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

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A137851 a(n) = A054525(n) * A061397(n).

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A156833 A054525 * A156348 * [1,2,3,...].

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A143519 Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)A010051(d); A054525 A010051.