A123709 -id:A123709 - cp's OEIS frontend

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

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

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

A123706 Matrix inverse of triangle A010766, where A010766(n,k) = [n/k], for n>=k>=1.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Oct 09 2006

Unsigned elements consist of only 0's, 1's and 2's.

			Triangle begins:
1;
-2, 1;
-1,-1, 1;
1,-1,-1, 1;
-1, 0, 0,-1, 1;
2, 0,-1, 0,-1, 1;
-1, 0, 0, 0, 0,-1, 1;
0, 0, 1,-1, 0, 0,-1, 1;
0, 1,-1, 0, 0, 0, 0,-1, 1;
2,-1, 0, 1,-1, 0, 0, 0,-1, 1;
-1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1;
-1, 1, 1,-1, 1,-1, 0, 0, 0, 0,-1, 1;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1;
2,-1, 0, 0, 0, 1,-1, 0, 0, 0, 0, 0,-1, 1;
1, 1,-1, 1,-1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1; ...

Enrique Pérez Herrero, Rows n = 1..100 of triangle, flattened

Cf. A102309, A085411; A123707, A123708, A123709.

t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0] - If[Divisible[n, k+1], MoebiusMu[n/(k+1)], 0]; Table[t[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 29 2013, after Enrique Pérez Herrero *)

T(n,k)=(matrix(n,n,r,c,r\c)^-1)[n,k]  \\ simplified by M. F. Hasler, Feb 12 2012

T(n,1) = +2 when n = 2*p where p is an odd prime.
T(n,1) = -2 when n is an even squarefree number with an odd number of prime divisors.
A123709(n) = number of nonzero terms in row n = 2^(m+1) - 1 when n is an odd number with exactly m distinct prime factors.
Sum_{k=1..n} T(n,k) = moebius(n).
Sum_{k=1..n} T(n,k)*k = 0 for n>1.
Sum_{k=1..n} T(n,k)*k^2 = 2*phi(n) for n>1 where phi(n)=A000010(n).
Sum_{k=1..n} T(n,k)*k^3 = 6*A102309(n) for n>1 where A102309(n)=Sum[d|n, moebius(d)*C(n/d,2) ].
Sum_{k=1..n} T(n,k)*k*2^(k-1) = A085411(n) = Sum_{d|n} mu(n/d)*(d+1)*2^(d-2) = total number of parts in all compositions of n into relatively prime parts.
T(n,k) = mu(n/k)-mu(n/(k+1)), where mu(n/k) is A008683(n/k) if k|n and 0 otherwise. - Enrique Pérez Herrero, Feb 21 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]))}

A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1.

Original entry on oeis.org

420, 630, 660, 780, 840, 924, 990, 1020, 1050, 1092, 1140, 1170, 1260, 1320, 1380, 1386, 1428, 1470, 1530, 1540, 1560, 1596, 1638, 1650, 1680, 1710, 1716, 1740, 1820, 1848, 1860, 1890, 1932, 1950, 1980, 2040, 2070, 2100, 2142, 2184, 2220, 2244

Offset: 1

M. F. Hasler, Feb 09 2012

nonn

I expect that A123709(a(k))=32.

M. F. Hasler, Table of n, a(n) for n = 1..18347

Cf. A200511, A178212.

Select[Range[2500], PrimeNu[#] == 4 && PrimeOmega[#] > 4 &](* Jean-François Alcover, Jun 30 2013 *)

is_A200521(n,c=4)={ omega(n)==c & bigomega(n)>c }

cp's OEIS Frontend

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

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

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

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A123706 Matrix inverse of triangle A010766, where A010766(n,k) = [n/k], for n>=k>=1.

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

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

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A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1.

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