A123706 -id:A123706 - cp's OEIS frontend

A123709 a(n) is the number of nonzero elements in row n of triangle A123706.

1, 2, 3, 4, 3, 4, 3, 4, 4, 6, 3, 8, 3, 6, 7, 4, 3, 8, 3, 8, 7, 6, 3, 8, 4, 6, 4, 8, 3, 11, 3, 4, 7, 6, 7, 8, 3, 6, 7, 8, 3, 11, 3, 8, 8, 6, 3, 8, 4, 8, 7, 8, 3, 8, 7, 8, 7, 6, 3, 16, 3, 6, 8, 4, 7, 12, 3, 8, 7, 14, 3, 8, 3, 6, 8, 8

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k]. a(n) = 4 when n is in A123710. a(n) = 8 when n is in A123711. a(n) = 16 when n is in A123712.

			a(n) = 3 when n is an odd prime.
a(n) = 7 when n is the product of two different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]
a(n) = 15 when n is the product of three different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]

M. F. Hasler, Table of n, a(n) for n = 1..500
Peter Luschny, Re: Is the A123706 triangle an extension of the Moebius function?, seqcomp list, Feb 12 2012

Cf. A123706, A123707, A123708, A123710, A123711, A123712; A010766.

Moebius[i_,j_]:=If[Divisible[i,j], MoebiusMu[i/j],0];
A123709[n_]:=Length[Select[Table[Moebius[n,j]-Moebius[n,j+1],{j,1,n}],#!=0&]];
Array[A123709, 500] (* Enrique Pérez Herrero, Feb 13 2012 *)

{a(n)=local(M=matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1); sum(k=1,n,if(M[n,k]==0,0,1))}

A123709(n)=#select((matrix(n, n, r, c, r\c)^-1)[n,],x->x)  \\ M. F. Hasler, Feb 12 2012

A123709(n)={ my(t=moebius(n)); sum(k=2,n, t+0 != t=if(n%k,0,moebius(n\k)))+1}  /* the "t+0 != ..." is required because of a bug in PARI versions <= 2.4.2, maybe beyond, which seems to be fixed in v. 2.5.1 */ \\ M. F. Hasler, Feb 13 2012

a(n) = 2^(m+1) - 1 when n is the product of m distinct odd primes. [Corrected by M. F. Hasler, Feb 13 2012]
For any k>1, a(n)=2^k if, and only if, n is a nonsquarefree number with A001221(n) = k-1 (= omega(n), number of distinct prime factors), with the only exception of a(n=6)=2^2. - M. F. Hasler, Feb 12 2012
A123709(n) = 1 + #{ k in 1..n-1 | Moebius(n,k+1) <> Moebius(n,k) }, where Moebius(n,k)={moebius(n/k) if n=0 (mod k), 0 else}, cf. link to message by P. Luschny. - M. F. Hasler, Feb 13 2012

A123711 Indices n such that A123709(n) = 8 = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224, 225

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].

It appears that this equals A200511, numbers of the form p^k q^m with k,m >= 1, k+m > 2 and p, q prime. - M. F. Hasler, Feb 12 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A123706, A123709, A123710, A123712; A010766.

Moebius[i_, j_] := If[Divisible[i, j], MoebiusMu[i/j], 0]; A123709[n_] :=
Length[Select[Table[Moebius[n, j] - Moebius[n, j + 1], {j, 1, n}], # != 0 &]]; Select[Range[500], A123709[#] == 8 &] (* G. C. Greubel, Apr 22 2017 *)

PARI

A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].
a(n) = A178212(n) for n <= 52, possibly more. [Reinhard Zumkeller, May 24 2010]
a(n) = A178212(n) for n <= 2000. - Bill McEachen, Jul 14 2024

G. C. Greubel, Table of n, a(n) for n = 1..1200

Cf. A123706, A123709, A123710, A123711, A010766.

Moebius[i_, j_] := If[Divisible[i, j], MoebiusMu[i/j], 0]; A123709[n_] := Length[Select[Table[Moebius[n, j] - Moebius[n, j + 1], {j, 1, n}], # != 0 &]]; Select[Range[6500], A123709[#] == 16 &] (* G. C. Greubel, Apr 22 2017 *)

is(n)=my(M=matrix(n, n, r, c,r\c)^-1); sum(k=1, n, M[n, k]!=0)==16 \\ Charles R Greathouse IV, Feb 09 2012

A123707 a(n) = Sum_{k=1..n} A123706(n,k)*2^(k-1).

Original entry on oeis.org

1, 0, 1, 3, 7, 14, 31, 60, 126, 248, 511, 1005, 2047, 4064, 8183, 16320, 32767, 65394, 131071, 261885, 524255, 1048064, 2097151, 4193220, 8388600, 16775168, 33554304, 67104765, 134217727, 268427002, 536870911, 1073725440, 2147483135

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766(n,k) = [n/k].

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A123706, A123708, A123709.

t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0] - If[Divisible[n, k + 1], MoebiusMu[n/(k + 1)], 0]; Table[Sum[t[n, k]*2^(k - 1), {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Oct 26 2017 *)

{a(n)=sum(k=1,n,(matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1)[n,k]*2^(k-1))}

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - x^k) / (1 - 2*x^k). - Ilya Gutkovskiy, Feb 06 2020

a(n) ~ 2^(n-2). - Vaclav Kotesovec, May 03 2025

A123708 a(n) = sum of unsigned elements in row n of triangle A123706.

Original entry on oeis.org

1, 3, 3, 4, 3, 5, 3, 4, 4, 7, 3, 8, 3, 7, 7, 4, 3, 8, 3, 8, 7, 7, 3, 8, 4, 7, 4, 8, 3, 13, 3, 4, 7, 7, 7, 8, 3, 7, 7, 8, 3, 13, 3, 8, 8, 7, 3, 8, 4, 8, 7, 8, 3, 8, 7, 8, 7, 7, 3, 16, 3, 7, 8, 4, 7, 13, 3, 8, 7, 15, 3, 8, 3, 7, 8, 8, 7, 13, 3, 8, 4, 7, 3, 16, 7, 7, 7, 8, 3, 16, 7, 8, 7, 7, 7, 8, 3, 8, 8, 8, 3, 13

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A123706, A123707, A123709; A010766.

t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0] - If[Divisible[n, k + 1], MoebiusMu[n/(k + 1)], 0]; Table[Sum[Abs[t[n, k]], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Oct 26 2017 *)

{a(n)=local(M=matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1);sum(k=1,n,abs(M[n,k]))}

A123710 Indices n such that 4 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

4, 6, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024

Offset: 1

Paul D. Hanna, Oct 09 2006

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].

Except for a(2)=6, these are proper prime powers, i.e., numbers p^k where k>1, p prime (A025475). - M. F. Hasler, Feb 12 2012

Cf. A123706, A123709, A123711, A123712; A010766.

for(n=1,1e4, A123709(n)==4 & print1(n","))  \\ M. F. Hasler, Feb 12 2012

a(n) = A025475(n) for n>2 (conjectured). - M. F. Hasler, Feb 12 2012

A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Dec 14 2021

We use the notation [n] = {0, 1, ..., n-1}. If S is a subset of [n] then we define the distset of S (set of distances of S) as {|x - y|: x, y in S}. We call a subset S of the natural numbers self-measuring if and only if S = distset(S).

			Triangle starts:
[ 0] [1]
[ 1] [1, 1]
[ 2] [1, 1,  1]
[ 3] [1, 1,  2, 1]
[ 4] [1, 1,  3, 1, 1]
[ 5] [1, 1,  4, 2, 1, 1]
[ 6] [1, 1,  5, 2, 1, 1, 1]
[ 7] [1, 1,  6, 3, 2, 1, 1, 1]
[ 8] [1, 1,  7, 3, 2, 1, 1, 1, 1]
[ 9] [1, 1,  8, 4, 2, 2, 1, 1, 1, 1]
[10] [1, 1,  9, 4, 3, 2, 1, 1, 1, 1, 1]
[11] [1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1]
[12] [1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1]
.
The first  column is 1,1,...  because {} = distset({}) and |{}| = 0.
The second column is 1,1,... because {0} = distset({0}) and |{0}| = 1.
The third  column is n-1  because {0, j} = distset({0, j}) and |{0, j}| = 2 for j = 1..n - 1.
The main diagonal is 1,1,... because [n] = distset([n]) and |[n]| = n (these are the complete rulers A103295).

Winston de Greef, Table of n, a(n) of the first 150 rows, flattened (n = 0..11324)
Peter Luschny, Illustrating self-measuring subsets of {0, 1, 2, 3}.

Cf. A350102 (row sums), A349976, A103295, A003988, A010766, A123706.

T := (n, k) -> ifelse(k < 2, 1, floor((n - 1) / (k - 1))):
seq(print(seq(T(n, k), k = 0..n)), n = 0..12);

distSet[s_] := Union[Map[Abs[Differences[#][[1]]] &, Union[Sort /@ Tuples[s, 2]]]]; T[n_, k_] := Count[Subsets[Range[0, n - 1]], ?((ds = distSet[#]) == # && Length[ds] == k &)]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Amiram Eldar, Dec 16 2021 *)

T(n, k) = if(k<=1, 1, (n - 1) \ (k - 1)) \\ Winston de Greef, Jan 31 2024

# generating and counting (slow)
def isSelfMeasuring(R):
    S, L = Set([]), len(R)
    R = Set([r - 1 for r in R])
    for i in range(L):
        for j in (0..i):
            S = S.union(Set([abs(R[i] - R[i - j])]))
    return R == S
def A349976row(n):
    counter = [0] * (n + 1)
    for S in Subsets(n):
        if isSelfMeasuring(S): counter[len(S)] += 1
    return counter
for n in range(10): print(A349976row(n))

T(n, k) = floor((n - 1) / (k - 1)) for k >= 2.

T(n, k) = 1 if k = 0 or k = 1 or n >= k >= floor((n + 1)/2).

A206706 Triangle read by rows, T(n,k) n>=0, 0<=k<=n; T(0,0) = -1 and for n > 0 T(n,k) = moebius(n,k+1) - moebius(n,k) where moebius(n,k) = mu(floor(n/k)) if k<>0 and k divides n, 0 otherwise; mu=A008683.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Feb 11 2012

This is a variant of Paul D. Hanna's A123706 which uses a definition given by Mats Granvik. It adds the column T(n,0) = mu(n) at the left hand side of the triangle.

The value T(0,0) was set to -1 to make the triangle invertible as a matrix with uniform signs of the entries of the inverse.

			[ 0]                    -1,
[ 1]                   1, -1,
[ 2]                 -1, 2, -1,
[ 3]                -1, 1, 1, -1,
[ 4]              0, -1, 1, 1, -1,
[ 5]             -1, 1, 0, 0, 1, -1,
[ 6]           1, -2, 0, 1, 0, 1, -1,
[ 7]          -1, 1, 0, 0, 0, 0, 1, -1,
[ 8]        0, 0, 0, -1, 1, 0, 0, 1, -1,
[ 9]       0, 0, -1, 1, 0, 0, 0, 0, 1, -1,
The inverse of this triangle as a matrix begins
[-1,  0,  0,  0,  0,  0,  0]
[-1, -1,  0,  0,  0,  0,  0]
[-1, -2, -1,  0,  0,  0,  0]
[-1, -3, -1, -1,  0,  0,  0]
[-1, -4, -2, -1, -1,  0,  0]
[-1, -5, -2, -1, -1, -1,  0]
[-1, -6, -3, -2, -1, -1, -1]

Cf. A123706, A008683.

with(numtheory): A206706 := proc(n,k) local moebius;
moebius := (n, k) -> `if`(k<>0 and irem(n,k) = 0, mobius(iquo(n,k)), 0);
moebius(n, k+1) - moebius(n, k) end:

mu[n_, k_] := If[k != 0 && Divisible[n, k], MoebiusMu[n/k], 0];
T[0, 0] = -1; T[n_, k_] /; 0 <= k <= n := mu[n, k+1] - mu[n, k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

def mur(n,k): return moebius(n//k) if k != 0 and n%k == 0 else 0
def A206706(n,k) : return -1 if n==0 and k==0 else mur(n,k+1) - mur(n,k)

cp's OEIS Frontend

A123709 a(n) is the number of nonzero elements in row n of triangle A123706.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A123711 Indices n such that A123709(n) = 8 = number of nonzero terms in row n of triangle A123706.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A123707 a(n) = Sum_{k=1..n} A123706(n,k)*2^(k-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A123708 a(n) = sum of unsigned elements in row n of triangle A123706.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A123710 Indices n such that 4 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

SageMath

Formula

A206706 Triangle read by rows, T(n,k) n>=0, 0<=k<=n; T(0,0) = -1 and for n > 0 T(n,k) = moebius(n,k+1) - moebius(n,k) where moebius(n,k) = mu(floor(n/k)) if k<>0 and k divides n, 0 otherwise; mu=A008683.