A007438 -id:A007438 - cp's OEIS frontend

A133727 A051731 * A007438 as a diagonalized matrix.

1, 1, 2, 1, 0, 5, 1, 2, 0, 7, 1, 0, 0, 0, 14, 1, 2, 5, 0, 0, 13, 1, 0, 0, 0, 0, 0, 27, 1, 2, 0, 7, 0, 0, 0, 26, 1, 0, 5, 0, 0, 0, 0, 0, 39, 1, 2, 0, 0, 14, 0, 0, 0, 0, 38

Offset: 1

Gary W. Adamson, Sep 21 2007

Row sums = the triangular numbers: (1, 3, 6, 10, 15, ...).

Right diagonal = A007438.

			First few rows of the triangle:
  1;
  1,  2;
  1,  0,  5;
  1,  2,  0,  7;
  1,  0,  0,  0, 14;
  1,  2,  5,  0,  0, 13;
  1,  0,  0,  0,  0,  0, 27;
  ...

Cf. A007438, A051731.

Inverse Mobius transform of A007438 as a diagonalized matrix; i.e., A051731 * (1; 0,2 0,0,5; 0,0,0,7; ...) where A007438 = (1, 2, 5, 7, 14, 13, 27, ...), the Mobius transform of the triangular numbers.

A015631 Number of ordered triples of integers from [ 1..n ] with no global factor.

Original entry on oeis.org

1, 3, 8, 15, 29, 42, 69, 95, 134, 172, 237, 287, 377, 452, 552, 652, 804, 915, 1104, 1252, 1450, 1635, 1910, 2106, 2416, 2674, 3007, 3301, 3735, 4027, 4522, 4914, 5404, 5844, 6432, 6870, 7572, 8121, 8805, 9389, 10249, 10831, 11776, 12506

Offset: 1

Olivier Gérard

nonn

Number of integer-sided triangles with at least two sides <= n and sides relatively prime. - Henry Bottomley, Sep 29 2006

			a(4) = 15 because the 15 triples in question are in lexicographic order: [1,1,1], [1,1,2], [1,1,3], [1,1,4], [1,2,2], [1,2,3], [1,2,4], [1,3,3], [1,3,4], [1,4,4], [2,2,3], [2,3,3], [2,3,4], [3,3,4] and [3,4,4]. - _Wolfdieter Lang_, Apr 04 2013
The a(4) = 15 triangles with at least two sides <= 4 and sides relatively prime (see _Henry Bottomley_'s comment above) are: [1,1,1], [1,2,2], [2,2,3], [1,3,3], [2,3,3], [2,3,4], [3,3,4], [3,3,5], [1,4,4], [2,4,5], [3,4,4], [3,4,5], [3,4,6], [4,4,5], [4,4,7]. - _Alois P. Heinz_, Feb 14 2020

R. J. Mathar, Table of n, a(n) for n = 1..10000

Column k=3 of A177976.

Cf. A000292, A002088, A015616, A015634, A015650, A134543.

[n eq 1 select 1 else Self(n-1)+ &+[MoebiusMu(n div d) *d*(d+1)/2:d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Feb 14 2020

with(numtheory):
b:= proc(n) option remember;
       add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
    end:
a:= proc(n) option remember;
      b(n) + `if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2011

a[1] = 1; a[n_] := a[n] = Sum[MoebiusMu[n/d]*d*(d+1)/2, {d, Divisors[n]}] + a[n-1]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jan 20 2014, after Maple *)
Accumulate[Table[Sum[MoebiusMu[n/d]*d*(d + 1)/2, {d, Divisors[n]}], {n, 1, 50}]] (* Vaclav Kotesovec, Jan 31 2019 *)

a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+1, 2))); \\ Seiichi Manyama, Jun 12 2021

a(n) = binomial(n+2, 3)-sum(k=2, n, a(n\k)); \\ Seiichi Manyama, Jun 12 2021

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^3)/(1-x)) \\ Seiichi Manyama, Jun 12 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A015631(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A015631(k1)
        j, k1 = j2, n//j2
    return n*(n-1)*(n+4)//6-c+j # Chai Wah Wu, Mar 30 2021

a(n) = (A071778(n)+3*A018805(n)+2)/6. - Vladeta Jovovic, Dec 01 2004
Partial sums of the Moebius transform of the triangular numbers (A007438). - Steve Butler, Apr 18 2006
a(n) = 2*A123324(n) - A046657(n) for n>1. - Henry Bottomley, Sep 29 2006
Row sums of triangle A134543. - Gary W. Adamson, Oct 31 2007
a(n) ~ n^3 / (6*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^3. - Ilya Gutkovskiy, Feb 14 2020
a(n) = n*(n+1)*(n+2)/6 - Sum_{j=2..n} a(floor(n/j)) = A000292(n) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

A117108 Moebius transform of tetrahedral numbers.

Original entry on oeis.org

1, 3, 9, 16, 34, 43, 83, 100, 155, 182, 285, 292, 454, 473, 636, 696, 968, 929, 1329, 1304, 1678, 1735, 2299, 2136, 2890, 2818, 3489, 3484, 4494, 4052, 5455, 5168, 6250, 6168, 7652, 6988, 9138, 8547, 10196, 9840, 12340, 10954, 14189, 13140, 15380, 14993, 18423

Offset: 1

Steve Butler, Apr 18 2006

nonn

Partial sums of a(n) give A015634(n).

See also A059358, A116963 (applied to shifted version of tetrahedral numbers), inverse Moebius transform of tetrahedral numbers. - Jonathan Vos Post, Apr 20 2006

			a(2) = 3 because of the triples (1,1,1), (1,1,2), (1,2,2).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000292 (tetrahedral numbers), A007438, A008683, A015634 (partial sums), A059358, A116963, A117109, A343544.

a[n_] := DivisorSum[n, MoebiusMu[n/#]*Binomial[# + 2, 3] &]; Array[a, 50] (* Amiram Eldar, Jun 07 2025 *)

a(n) = sumdiv(n, d, binomial(d+2, 3)*moebius(n/d)); \\ Michel Marcus, Nov 04 2018

a(n) = |{(x,y,z) : 1 <= x <= y <= z <= n, gcd(x,y,z,n) = 1}|.

G.f.: Sum_{k>=1} mu(k) * x^k / (1 - x^k)^4. - Ilya Gutkovskiy, Feb 13 2020

Offset changed to 1 by Ilya Gutkovskiy, Feb 13 2020

A117109 Moebius transform of binomial(n+3, 4).

Original entry on oeis.org

1, 4, 14, 30, 69, 107, 209, 295, 480, 641, 1000, 1209, 1819, 2166, 2976, 3546, 4844, 5379, 7314, 8110, 10402, 11645, 14949, 15890, 20405, 21927, 26910, 29055, 35959, 37108, 46375, 48484, 57890, 61196, 73536, 75027, 91389, 93951, 110096, 114260

Offset: 1

Steve Butler, Apr 18 2006

nonn

Partial sums of a(n) give A015650(n).

			a(2)=4 because of the quadruples (1,1,1,1), (1,1,1,2), (1,1,2,2), (1,2,2,2).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007438, A015650, A117108.

b34:= unapply(expand(binomial(n+3,4)),n):
f:= proc(n) local k; uses numtheory;
add(b34(k)*mobius(n/k),k=divisors(n))
end proc:
map(f, [$1..100]); # Robert Israel, May 24 2019

a[n_] := Sum[Binomial[k+3, 4] MoebiusMu[n/k], {k, Divisors[n]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 01 2023 *)

a(n) = sumdiv(n, k, binomial(k+3, 4)*moebius(n/k)); \\ Michel Marcus, Nov 04 2018

a(n) = |{(x,y,z,w) : 1 <= x <= y <= z <= w <= n, gcd(x,y,z,w,n) = 1}|.

G.f.: Sum_{k>=1} mu(k) * x^k / (1 - x^k)^5. - Ilya Gutkovskiy, Feb 13 2020

A332470 a(n) = Sum_{d|n} mu(n/d) * binomial(n+d-2, n-1).

Original entry on oeis.org

1, 1, 5, 16, 69, 226, 923, 3312, 12825, 47896, 184755, 700712, 2704155, 10373455, 40113421, 154946976, 601080389, 2332498482, 9075135299, 35338355380, 137846298360, 538213522254, 2104098963719, 8233142596640, 32247603662625, 126408753954731, 495918514791900

Offset: 1

Ilya Gutkovskiy, Feb 13 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000010, A007438, A008683, A117108, A117109, A332508, A343547.

[&+[MoebiusMu(n div d) *Binomial(n+d-2,n-1):d in Divisors(n)]:n in [1..30]]; // Marius A. Burtea, Feb 13 2020

Table[DivisorSum[n, MoebiusMu[n/#] Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[MoebiusMu[k] x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

a(n) = [x^n] Sum_{k>=1} mu(k) * x^k / (1 - x^k)^n.

a(n) = |{(x_1, x_2, ... , x_{n-1}) : 1 <= x_1 <= x_2 <= ... <= x_n = n, gcd(x_1, x_2, ... , x_n) = 1}|. - Seiichi Manyama, Apr 20 2021

A030050 Numbers from the Conway-Schneeberger 15-theorem.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 14, 15

Offset: 1

N. J. A. Sloane

The 15-theorem asserts that a positive definite integral quadratic form represents all numbers iff it represents the numbers in this sequence. "Integral" here means that the quadratic form equals x^T M x, where x is an integer vector and M is an integer matrix. - T. D. Noe, Mar 30 2006

Union of the first five triangular numbers {1, 3, 6, 10, 15} and their Möbius transform {1, 2, 5, 7, 14}, in ascending order. - Daniel Forgues, Feb 24 2015

			a(2*1) = Sum{d|(1+1)} mu(d) t_{(1+1)/d} = 1 * t_2 + (-1) * t_1 = 3 - 1 = 2;
a(2*2) = Sum{d|(2+1)} mu(d) t_{(2+1)/d} = 1 * t_3 + (-1) * t_1 = 6 - 1 = 5;
a(2*3) = Sum{d|(3+1)} mu(d) t_{(3+1)/d} = 1 * t_4 + (-1) * t_2 + 0 * t_1 = 10 - 3 = 7;
a(2*4) = Sum{d|(4+1)} mu(d) t_{(4+1)/d} = 1 * t_5 + (-1) * t_1 = 15 - 1 = 14.

Manjul Bhargava, On the Conway-Schneeberger fifteen theorem, Contemporary Mathematics 272 (1999), 27-37.
J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 141.
J. H. Conway, Universal quadratic forms and the fifteen theorem, Contemporary Mathematics 272 (1999), 23-26.
J. H. Conway and W. A. Schneeberger, personal communication.

Manjul Bhargava, The Fifteen Theorem and Generalizations
Ivars Peterson, All Square: Science News Online (subscription required)
Wikipedia, 15 and 290 theorems.

Cf. A030051, A116582, A154363.

a[n_] := If[OddQ[n], (n+1)*(n+3)/8, DivisorSum[n/2+1, MoebiusMu[#]*(n+2#+2)*(n+2)/(8#^2) &]]; Array[a, 9] (* Jean-François Alcover, Dec 03 2015 *)

From Daniel Forgues, Feb 24 & 26 2015: (Start)
a(2n-1) = t_n = n*(n+1)/2 = A000217(n), 1 <= n <= 5;
a(2n) = Sum{d|(n+1)} mu(d) t_{(n+1)/d} = A007438(n+1), 1 <= n <= 4. (End)

A159905 Triangle read by rows, Mobius transform of A004736.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Apr 25 2009

Row sums = A007438, Mobius transform of the triangular numbers.

Left border = phi(n), A000010.

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

Cf. A007438

Triangle read by rows, A054525 * A004736. A054525 = the Mobius transform, A004736 = the natural number "decrescendo" triangle: (1; 2,1; 3,2,1;...)

A177975 Square array T(n,k) read by antidiagonals up. Each column is the first column in the matrix inverse of a triangular matrix that is the k-th differences of A051731 in the column direction.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 7, 9, 4, 1, 0, 2, 14, 16, 14, 5, 1, 0, 6, 13, 34, 30, 20, 6, 1, 0, 4, 27, 43, 69, 50, 27, 7, 1, 0, 6, 26, 83, 107, 125, 77, 35, 8, 1, 0, 4, 39, 100, 209, 226, 209, 112, 44, 9, 1, 0, 10, 38, 155, 295, 461, 428, 329, 156, 54, 10, 1

Offset: 1

Mats Granvik, May 16 2010

The first column in this array is the first column in A134540 which is the matrix inverse of A177978. The second column is the first column in A159905. The rows in this array are described by both binomial expressions and closed form polynomials.

			Table begins:
  1..1...1...1....1.....1.....1......1......1.......1.......1
  0..1...2...3....4.....5.....6......7......8.......9......10
  0..2...5...9...14....20....27.....35.....44......54......65
  0..2...7..16...30....50....77....112....156.....210.....275
  0..4..14..34...69...125...209....329....494.....714....1000
  0..2..13..43..107...226...428....749...1234....1938....2927
  0..6..27..83..209...461...923...1715...3002....5004....8007
  0..4..26.100..295...736..1632...3312...6270...11220...19162
  0..6..39.155..480..1266..2975...6399..12825...24255...43692
  0..4..38.182..641..1871..4789..11103..23807...47896...91367
  0.10..65.285.1000..3002..8007..19447..43757...92377..184755
  0..4..50.292.1209..4066.11837..30920..74139..165748..349438
  0.12..90.454.1819..6187.18563..50387.125969..293929..646645
  0..6..75.473.2166..8101.26202..75797.200479..492406.1136048
  0..8.100.636.2976.11482.38523.115915.319231..816421.1960190
  0..8.100.696.3546.14712.52548.167112.483879.1296064.3249312

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Column k=1..5 gives A063524, A000010, A007438, A117108, A117109.
Main diagonal gives A332470.
Cf. A177976, A177977.

T(n, k) = sumdiv(n, d, moebius(n/d)*binomial(d+k-2, d-1)); \\ Seiichi Manyama, Jun 12 2021

From Seiichi Manyama, Jun 12 2021: (Start)
G.f. of column k: Sum_{j>=1} mu(j) * x^j/(1 - x^j)^k.
T(n,k) = Sum_{d|n} mu(n/d) * binomial(d+k-2,d-1). (End)

A343565 a(n) = |{(x_1, x_2, ... , x_n) : 1 <= x_1 <= x_2 <= ... <= x_n <= n, gcd(x_1, x_2, ... , x_n, n) = 1}|.

Original entry on oeis.org

1, 2, 9, 30, 125, 428, 1715, 6270, 24255, 91367, 352715, 1345448, 5200299, 20019526, 77554749, 300295038, 1166803109, 4535971916, 17672631899, 68913247655, 269128640958, 1051984969598, 4116715363799, 16123381989000, 63205303195125, 247956558998878, 973469689288236

Offset: 1

Seiichi Manyama, Apr 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000010, A007438, A117108, A117109, A332470, A343517, A343548, A343549.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[# + n - 1, n] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)

a(n) = sumdiv(n, d, moebius(n/d)*binomial(d+n-1, n));

a(n) = Sum_{d|n} mu(n/d) * binomial(d+n-1, n).

a(n) = [x^n] Sum_{k>=1} mu(k) * x^k/(1 - x^k)^(n+1).

A133725 a(n) = Sum_{d|n} mu(n/d)d(3*d - 1)/2.

Original entry on oeis.org

1, 4, 11, 17, 34, 35, 69, 70, 105, 106, 175, 142, 246, 213, 284, 284, 424, 321, 531, 428, 570, 535, 781, 572, 890, 750, 963, 858, 1246, 860, 1425, 1144, 1430, 1288, 1716, 1290, 2034, 1611, 2004, 1720, 2500, 1722, 2751, 2150, 2580, 2365, 3289, 2296, 3507, 2690

Offset: 1

Gary W. Adamson, Sep 21 2007

nonn

Previous name was: A054525 * A000326.

Möbius transform of the pentagonal numbers.

			a(4) = 17 = (0, -1, 0, 1) dot (1, 5, 12, 22) = (0, -5, 0, 22).

Cf. A000010, A000326, A007434, A007438, A008683 (mu), A054525.

read("transforms") : A000326 := proc(n) n*(3*n-1)/2 ; end: a000326 := [seq(A000326(n),n=1..300)] ; a133725 := MOBIUS(a000326) ; for i from 1 to nops(a133725) do printf("%d,",op(i,a133725)) ; od: # R. J. Mathar, Jan 19 2009

a[n_] := DivisorSum[n, #*(3*#-1) * MoebiusMu[n/#] &] / 2; Array[a, 50] (* Amiram Eldar, May 29 2025 *)

a(n) = sumdiv(n, d, d*(3*d-1) * moebius(n/d)) / 2; \\ Amiram Eldar, May 29 2025

G.f.: Sum_{k>=1} mu(k) * x^k * (1 + 2*x^k) / (1 - x^k)^3. - Ilya Gutkovskiy, Sep 17 2021

a(n) = (3*A007434(n) - A000010(n))/2. - Amiram Eldar, Jun 04 2025

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

New name from Ilya Gutkovskiy, Sep 17 2021

cp's OEIS Frontend

A133727 A051731 * A007438 as a diagonalized matrix.

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A015631 Number of ordered triples of integers from [ 1..n ] with no global factor.

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A117108 Moebius transform of tetrahedral numbers.

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A117109 Moebius transform of binomial(n+3, 4).

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A332470 a(n) = Sum_{d|n} mu(n/d) * binomial(n+d-2, n-1).

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A030050 Numbers from the Conway-Schneeberger 15-theorem.

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A159905 Triangle read by rows, Mobius transform of A004736.

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A133725 a(n) = Sum_{d|n} mu(n/d)d(3*d - 1)/2.