cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-46 of 46 results.

A346760 a(n) = Sum_{d|n} mu(n/d) * binomial(d,3).

Original entry on oeis.org

0, 0, 1, 4, 10, 19, 35, 52, 83, 110, 165, 196, 286, 329, 444, 504, 680, 713, 969, 1016, 1294, 1375, 1771, 1752, 2290, 2314, 2841, 2908, 3654, 3476, 4495, 4400, 5290, 5304, 6500, 6124, 7770, 7467, 8852, 8688, 10660, 9802, 12341, 11700, 13652, 13409, 16215, 14768, 18389, 17190
Offset: 1

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Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Crossrefs

3rd column of A020921.
Cf. A000010, A000292, A007434, A059376, A102309, A117108, A346761.

Programs

  • Mathematica
    Table[Sum[MoebiusMu[n/d] Binomial[d, 3], {d, Divisors[n]}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[Sum[MoebiusMu[k] x^(3 k)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*binomial(d, 3)); \\ Michel Marcus, Aug 03 2021

Formula

G.f.: Sum_{k>=1} mu(k) * x^(3*k) / (1 - x^k)^4.
a(n) = (A059376(n) - 3 * A007434(n) + 2 * A000010(n)) / 6.

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

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Crossrefs

4th column of A020921.
Cf. A000010, A000332, A007434, A059376, A059377, A102309, A117109, A346760.

Programs

  • Mathematica
    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

Formula

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.

A192000 Sum of binomial numbers A000332(k+3), with k in the reduced residue system modulo n.

Original entry on oeis.org

0, 1, 6, 16, 56, 71, 252, 296, 651, 721, 2002, 1282, 4368, 3402, 5782, 6672, 15504, 7947, 26334, 15702, 28868, 28457, 65780, 30212, 85580, 63063, 103284, 81452, 201376, 66102, 278256, 174624, 255794, 228684, 383166, 206838, 658008, 391419, 576394, 413244, 1086008
Offset: 1

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Author

Wolfdieter Lang, Jun 22 2011

Keywords

Comments

The reduced residue system modulo n used here is the set of numbers k from the set {0,1,...,n-1} which satisfy gcd(k,n)=1. There are phi(n) = A000010(n) such numbers k.
This is the m=4 member of a family of sequences, call them rmnS(m) (reduced mod n sum), with entries rmnS(m;n):=sum(binomial(k+m-1,m),0<=k<=n-1 with gcd(k,n)=1), m>=0, n>=1. Recall gcd(0,n)=n.
The members for m=0, 1, 2 and 3 are A000010, A023896, A127415, and A189918, respectively, where in the m=1 and 2 cases the offset for n=1 should be taken as 0 (not 1).

Examples

			a(6) = A000332(4) + A000292(8)= 1 + 70 = 71.
a(6) = (6/6!)*(6*3666*(1/3) + 5*137*2 - 182) = 71.
a(12) = A000332(4) + A000332(8) + A000332(10) + A000332(14) = 1 + 70 + 210 + 1001 = 1282.
a(12) = (12/6!)*(12*18258*(1/3) + 5*407*2 - 182) = 1282.

Crossrefs

Cf. A189918, A023896, A127415, A189922.

Programs

  • PARI
    a(n) = sum(k=0, n-1, if (gcd(n,k) == 1, binomial(k+3, 4))); \\ Michel Marcus, Feb 01 2016

Formula

a(n) = sum(A000332(k+3), 0<=k<=n-1, gcd(k,n)=1), n>=1.
a(n) = (n/6!)*(n*(6*n^3+45*n^2+110*n+90)*P(1,n) + 5*(2*n^2+9*n+11)*P(-1,n) - P(-3,n)), n>=2, with P(k,n):= J(k,n)/n^k, where J(k,n) is the Jordan function (see A000010, A007434, A059376 - A059378, A069091 - A069095).

Extensions

More terms from Michel Marcus, Feb 01 2016

A332617 a(n) = Sum_{k=1..n} J_n(k), where J is the Jordan function, J_n(k) = k^n * Product_{p|k, p prime} (1 - 1/p^n).

Original entry on oeis.org

1, 4, 34, 336, 4390, 66312, 1197858, 24612000, 574002448, 14903406552, 427622607366, 13419501812640, 457579466056498, 16840326075104280, 665473192580864556, 28101209228393371200, 1262896789586657015796, 60182268296582518426368, 3031282541337682050032664
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Links

Crossrefs

Cf. A000010, A002088, A007434, A059376, A059377, A059378, A059379, A059380, A067858, A319194, A321879.

Programs

  • Magma
    [&+[&+[MoebiusMu(k div d)*d^n:d in Divisors(k)]:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[Sum[Sum[MoebiusMu[k/d] d^n, {d, Divisors[k]}], {k, 1, n}], {n, 1, 19}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[Sum[MoebiusMu[k] j^n x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]

Formula

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

A336488 Values taken by all the Jordan totient functions J_k(m) for k >= 1 and m >= 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 20, 22, 24, 26, 28, 30, 31, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 63, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 124, 126, 127, 128, 130, 132, 136, 138, 140, 144, 148
Offset: 1

Views

Author

Amiram Eldar, Jul 23 2020

Keywords

Comments

The asymptotic density of this sequence is 0 (Rao and Murty, 1979).
First differs from A221178 at n = 75, since a(75) = J_3(6) = 182 is not a term of A221178.

Links

  • R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.

Crossrefs

A002202 is a subsequence.
Cf. A000010, A007434, A059376, A059377, A059378, A059379, A059380, A069091, A069092, A069093, A069094, A069095, A221178.
Similar sequence: A211347.

Programs

  • Mathematica
    phiQ[m_] := Select[Range[m + 1, 2 m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; jor[k_, n_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; jorval[k_, mx_] := jor[k, #] & /@ Range[Floor@Surd[mx*Zeta[k], k]]; mx = 300; Select[Union @ Flatten[{Select[Range[mx], phiQ], jorval[#, mx] & /@ Range[2, Floor[Log2[mx]]]}], # <= mx &] (* using code by Jean-François Alcover at A002202 *)

A369101 Dirichlet g.f.: zeta(s-3)^2 * (1 - 2^(4-s)) / zeta(s).

Original entry on oeis.org

1, -1, 53, -64, 249, -53, 685, -960, 2133, -249, 2661, -3392, 4393, -685, 13197, -11264, 9825, -2133, 13717, -15936, 36305, -2661, 24333, -50880, 46625, -4393, 76545, -43840, 48777, -13197, 59581, -118784, 141033, -9825, 170565, -136512, 101305, -13717, 232829
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 13 2024

Keywords

Comments

In general, for k > 0, if Dirichlet g.f. is zeta(s-k)^2 * (1 - 2^(k+1-s)) / zeta(s), then a(n) ~ log(2) * n^(k+1) / ((k+1) * zeta(k+1)).

Crossrefs

Cf. A048272 (k=0), A332794 (k=1), A368929 (k=2).
Cf. A000578, A059376.

Programs

  • Mathematica
    Table[Sum[DivisorSum[k, #^3*MoebiusMu[k/#]&]*(-1)^(n/k+1)*(n/k)^3, {k, Divisors[n]}], {n, 1, 50}]
f[p_, e_] := p^(3*e-3) * (1 + (e+1)*(p^3-1)); f[2, e_] := -(7*e-6)*8^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2024 *)
  • PARI
    a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e=f[i,2]; if(p == 2, -(7*e-6)*8^(e-1), p^(3*e-3) * (1 + (e+1)*(p^3-1))));} \\ Amiram Eldar, Jan 13 2024

Formula

Sum_{k=1..n} a(k) ~ 45 * log(2) * n^4 / (2*Pi^4).
Multiplicative with a(2^e) = -(7*e-6)*8^(e-1), and a(p^e) = p^(3*e-3) * (1 + (e+1)*(p^3-1)) for an odd prime p. - Amiram Eldar, Jan 13 2024
Previous Showing 41-46 of 46 results.

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