A015634 -id:A015634 - cp's OEIS frontend

A082540 Number of ordered quadruples (a,b,c,d) with gcd(a,b,c,d)=1 (1 <= {a,b,c,d} <= n).

1, 15, 79, 239, 607, 1199, 2303, 3823, 6223, 9279, 13919, 19183, 27007, 35743, 47519, 60735, 78719, 97103, 122447, 148527, 181839, 216959, 262543, 306863, 365343, 423855, 495855, 569055, 661679, 748527, 862047, 972191, 1104831, 1237247

Offset: 1

Benoit Cloitre, May 11 2003

nonn

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Column k=4 of A344527.
Cf. A018805, A071778, A082544, A343978.
Cf. A015634.

a(n)=sum(k=1,n,moebius(k)*floor(n/k)^4)

from functools import lru_cache
@lru_cache(maxsize=None)
def A082540(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082540(k1)
        j, k1 = j2, n//j2
    return n*(n**3-1)-c+j # Chai Wah Wu, Mar 29 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^4.
a(n) is asymptotic to c*n^4 with c=0.92393....
Lim_{n->infinity} a(n)/n^4 = 1/zeta(4) = A215267 = 90/Pi^4. - Karl-Heinz Hofmann, Apr 11 2021
Lim_{n->infinity} n^4/a(n) =   zeta(4) = A013662 = Pi^4/90. - Karl-Heinz Hofmann, Apr 11 2021
a(n) = n^4 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024

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

A177976 Square array T(n,k) read by antidiagonals up. Cumulative column sums of A177975.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 10, 15, 13, 5, 1, 1, 12, 29, 29, 19, 6, 1, 1, 18, 42, 63, 49, 26, 7, 1, 1, 22, 69, 106, 118, 76, 34, 8, 1, 1, 28, 95, 189, 225, 201, 111, 43, 9, 1, 1, 32, 134, 289, 434, 427, 320, 155, 53, 10, 1, 1, 42, 172, 444, 729, 888, 748, 484, 209, 64, 11, 1

Offset: 1

Mats Granvik, May 16 2010

Each row is described by both a binomial expression and a closed form polynomial. The closed form polynomials given in A177977 extends this table to the left. For example the 0th column is A002321 and the -1st column is A092149.

Also number of ordered k-tuples of integers from [ 1..n ] with no global factor. - Seiichi Manyama, Jun 12 2021

			Table begins:
  1..1...1....1.....1.....1......1......1.......1.......1.......1
  1..2...3....4.....5.....6......7......8.......9......10......11
  1..4...8...13....19....26.....34.....43......53......64......76
  1..6..15...29....49....76....111....155.....209.....274.....351
  1.10..29...63...118...201....320....484.....703.....988....1351
  1.12..42..106...225...427....748...1233....1937....2926....4278
  1.18..69..189...434...888...1671...2948....4939....7930...12285
  1.22..95..289...729..1624...3303...6260...11209...19150...31447
  1.28.134..444..1209..2890...6278..12659...24034...43405...75139
  1.32.172..626..1850..4761..11067..23762...47841...91301..166506
  1.42.237..911..2850..7763..19074..43209...91598..183678..351261
  1.46.287.1203..4059.11829..30911..74129..165737..349426..700699
  1.58.377.1657..5878.18016..49474.124516..291706..643355.1347344
  1.64.452.2130..8044.26117..75676.200313..492185.1135761.2483392
  1.72.552.2766.11020.37599.114199.316228..811416.1952182.4443582
  1.80.652.3462.14566.52311.166747.483340.1295295.3248246.7692894

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

Column k=1..5 gives A000012, A002088, A015631, A015634, A015650.

Cf. A177975, A177977, A344527, A345229.

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

T(n, k) = binomial(n+k-1, k)-sum(j=2, n, T(n\j, k)); \\ Seiichi Manyama, Jun 12 2021

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

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

A177977 Triangle read by rows. Polynomials based on sums of Moebius transforms.

Original entry on oeis.org

1, 1, 0, 1, 3, -2, 1, 6, 5, -6, 1, 10, 35, 26, -48, 1, 15, 85, 165, -26, -120, 1, 21, 175, 735, 1264, -36, -1440, 1, 28, 322, 1960, 5929, 8092, -1212, -10080, 1, 36, 546, 4536, 22449, 60564, 57644, -24816, -80640, 1, 45, 870, 9450, 63273, 254205, 572480

Offset: 1

Mats Granvik, May 16 2010

These polynomials were found by entering the rows of A177976 in Wolfram Alpha. The lower left half equals part of the Stirling numbers of the first kind given in table A094638. To evaluate, enter a value for n and divide row sums with factorial numbers as shown in the example section. n=-1 gives A092149, n=0 gives the Mertens function A002321, n=1 gives A000012, n=2 gives A002088, n=3 gives A015631, and n=4 gives A015634.

			Triangle begins and the polynomials are:
(1*n^0)/1
(1*n^1 +0*n^0)/1
(1*n^2 +3*n^1 -2*n^0)/2
(1*n^3 +6*n^2 +5*n^1 -6*n^0)/6
(1*n^4 +10*n^3 +35*n^2 +26*n^1 -48*n^0)/24
(1*n^5 +15*n^4 +85*n^3 +165*n^2 -26*n^1 -120*n^0)/120
(1*n^6 +21*n^5 +175*n^4 +735*n^3 +1264*n^2 -36*n^1 -1440*n^0)/720

Typo in sequence (erroneous comma) corrected by N. J. A. Sloane, May 22 2010

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A082540 Number of ordered quadruples (a,b,c,d) with gcd(a,b,c,d)=1 (1 <= {a,b,c,d} <= n).

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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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A177976 Square array T(n,k) read by antidiagonals up. Cumulative column sums of A177975.

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

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A177977 Triangle read by rows. Polynomials based on sums of Moebius transforms.

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