A117109 -id:A117109 - cp's OEIS frontend

A117108 Moebius transform of tetrahedral numbers.

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

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

A015650 Number of ordered 5-tuples of integers from [ 1..n ] with no global factor.

Original entry on oeis.org

1, 5, 19, 49, 118, 225, 434, 729, 1209, 1850, 2850, 4059, 5878, 8044, 11020, 14566, 19410, 24789, 32103, 40213, 50615, 62260, 77209, 93099, 113504, 135431, 162341, 191396, 227355, 264463, 310838, 359322, 417212, 478408, 551944, 626971, 718360, 812311, 922407, 1036667

Offset: 1

Olivier Gérard

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Column k=5 of A177976.
Partial sums of A117109.
Cf. A000389, A002088, A013663, A015631, A015634.

a[n_] := Sum[DivisorSum[k, MoebiusMu[k/#]*Binomial[# + 3, 4] &], {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jun 07 2025 *)

a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+3, 4))); \\ Seiichi Manyama, Jun 12 2021

a(n) = binomial(n+4, 5)-sum(k=2, n, a(n\k)); \\ Seiichi Manyama, Jun 12 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^5)/(1-x)) \\ Seiichi Manyama, Jun 12 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A015650(n):
    if n == 0:
        return 0
    c, j = n+1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A015650(k1)
        j, k1 = j2, n//j2
    return n*(n+1)*(n+2)*(n+3)*(n+4)//120-c+j # Chai Wah Wu, Apr 18 2021

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^5. - Ilya Gutkovskiy, Feb 14 2020
a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/120 - Sum_{j=2..n} a(floor(n/j)) = A000389(n+4) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Apr 18 2021
a(n) ~ n^5 / (120*zeta(5)). - Amiram Eldar, Jun 08 2025

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).

A346761 a(n) = Sum_{d|n} mu(n/d) * binomial(d,4).

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 69, 126, 205, 330, 479, 715, 966, 1360, 1750, 2380, 2919, 3876, 4634, 5950, 6985, 8855, 10062, 12645, 14235, 17424, 19473, 23751, 25820, 31465, 34140, 40590, 43996, 52320, 55365, 66045, 69939, 81536, 86476, 101270, 104964, 123410, 128435, 147504

Offset: 1

Ilya Gutkovskiy, Aug 02 2021

nonn

4th column of A020921.

Cf. A000010, A000332, A007434, A059376, A059377, A102309, A117109, A346760.

Table[Sum[MoebiusMu[n/d] Binomial[d, 4], {d, Divisors[n]}], {n, 1, 45}]
nmax = 45; CoefficientList[Series[Sum[MoebiusMu[k] x^(4 k)/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} mu(k) * x^(4*k) / (1 - x^k)^5.

a(n) = (A059377(n) - 6 * A059376(n) + 11 * A007434(n) - 6 * A000010(n)) / 24.

cp's OEIS Frontend

A117108 Moebius transform of tetrahedral numbers.

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

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A015650 Number of ordered 5-tuples of integers from [ 1..n ] with no global factor.

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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.

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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}|.

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A346761 a(n) = Sum_{d|n} mu(n/d) * binomial(d,4).

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