A059377 -id:A059377 - cp's OEIS frontend

A255819 E.g.f.: exp(Sum_{k>=1} k^3 * x^k).

1, 1, 17, 211, 3049, 54221, 1131601, 26714647, 700868561, 20208794329, 634445325361, 21512122643771, 782497124407417, 30364699568650981, 1251108918727992689, 54512805637285532671, 2502891521610396838561, 120718449425308259052977, 6099522639316776103853521

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if e.g.f. = exp(Sum_{k>=1} k^m * x^k) and m>0, then a(n) ~ (m+2)^(-1/2) * Gamma(m+2)^(1/(2*m+4)) * exp((m+2)/(m+1) * Gamma(m+2)^(1/(m+2)) * n^((m+1)/(m+2)) + zeta(-m) - n) * n^(n - 1/(2*m+4)).
It appears that the sequence a(n) taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k: if true, then the sequence a(n) taken modulo k would be periodic with period dividing k. - Peter Bala, Nov 14 2017
The above conjecture is true - see the Bala link. - Peter Bala, Jan 20 2018

G. C. Greubel, Table of n, a(n) for n = 0..250
P. Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A082579, A255807, A000262.

nmax=20; CoefficientList[Series[Exp[Sum[k^3*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k^3*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)

E.g.f.: exp(x*(1 + 4*x + x^2)/(1-x)^4).
a(n) ~ 2^(3/10) * 3^(1/10) * 5^(-1/2) * n^(n-1/10) * exp(1/120 + 5 * 2^(-7/5) * 3^(1/5) * n^(4/5) - n).
a(n) = y(n)*n! where y(0)=1 and y(n)=(Sum_{k=0..n-1} (n-k)^4*y(k))/n for n>=1. - Benedict W. J. Irwin, Jun 02 2016
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_4(k)/k), where J_4(k) is the Jordan function (A059377). - Ilya Gutkovskiy, May 25 2019

A160891 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5.

Original entry on oeis.org

1, 15, 40, 120, 156, 600, 400, 960, 1080, 2340, 1464, 4800, 2380, 6000, 6240, 7680, 5220, 16200, 7240, 18720, 16000, 21960, 12720, 38400, 19500, 35700, 29160, 48000, 25260, 93600, 30784, 61440, 58560, 78300, 62400, 129600, 52060, 108600, 95200

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^4 such that the quotient group Z^4 / L is C_nm x (C_m)^3 (and also (C_nm)^3 x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

			There are 1395 = A160870(8,4) lattices of volume 8 in Z^4. Among them, a(8) = 960 give the quotient group C_8 and a(2) = 15 give C_2 x C_2 x C_2.
Among the lattices of volume 64 in Z^4, there are a(4) = 120 such that the quotient group is C_4 x C_4 x C_4 and other 120 with quotient group C_8 x (C_2)^3.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
Index to Jordan function ratios J_k/J_1.

Column 4 of A263950.

Cf. A000010, A000203, A059377, A160870.

A160891 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(3*e-3)*(1+p+p^2+p^3) ; end do; a; end proc:
seq(A160891(n),n=1..20) ; # R. J. Mathar, Jul 10 2011

A160891[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(5-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 19 2010 *)
f[p_, e_] := p^(3 e - 3)*(1 + p + p^2 + p^3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 08 2022 *)

vector(50, n, sumdiv(n^3, d, if(ispower(d, 4), moebius(sqrtnint(d, 4))*sigma(n^3/d), 0))) \\ Altug Alkan, Oct 30 2014

a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(3*f[i,2]-3)*(1+p+p^2+p^3); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_4(n)/J_1(n) = J_4(n)/phi(n) = A059377(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 19 2010
Multiplicative with a(p^e) = p^(3e-3)*(1+p+p^2+p^3). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^3). - Álvar Ibeas, Oct 30 2015
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/((p^3-1)*(p^3+p^2+p+1))) = 1.115923965261131974852254388404911045036763705978837384729819264463715993... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.4629765396... . - Amiram Eldar, Nov 08 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^4). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Aug 22 2010

A338549 a(n) = n^4 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^4.

Original entry on oeis.org

1, -17, 80, -240, 624, -1360, 2400, -3840, 6480, -10608, 14640, -19200, 28560, -40800, 49920, -61440, 83520, -110160, 130320, -149760, 192000, -248880, 279840, -307200, 390000, -485520, 524880, -576000, 707280, -848640, 923520, -983040, 1171200, -1419840, 1497600, -1555200

Offset: 1

Ilya Gutkovskiy, Nov 02 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A059377, A279395, A325596, A338547, A338548.

Table[n^4 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^4, {d, Divisors[n]}], {n, 1, 36}]
nmax = 36; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - 11 x^k + 11 x^(2 k) - x^(3 k))/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^4 - 1)*p^(4*(e - 1)); f[2, 1] = -17; f[2, e_] := -15*2^(4*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = n^4 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^4); \\ Michel Marcus, Nov 02 2020

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - 11*x^k + 11*x^(2*k) - x^(3*k)) / (1 + x^k)^5.
G.f. A(x) satisfies: A(x) = x * (1 - 11*x + 11*x^2 - x^3) / (1 + x)^5 - Sum_{k>=2} A(x^k).
Dirichlet g.f.: (1 - 2^(5 - s)) * zeta(s - 4) / zeta(s).
a(n) = J_4(n) if n odd, J_4(n) - 32 * J_4(n/2) if n even, where J_4 = A059377 (Jordan function J_4).
Multiplicative with a(2) = -17, a(2^e) = -15*2^(4*(e-1)) for e > 1, and a(p^e) = (p^4-1)*p^(4*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).

Original entry on oeis.org

1, 466, 9681, 123106, 487345, 4511346, 6722401, 31576546, 63779361, 227102770, 235777201, 1191789186, 883674961, 3132638866, 4717986945, 8084578786, 7411648321, 29721182226, 17926949521, 59995093570, 65079564081, 109872175666, 81870270241, 305692541826

Offset: 1

Seiichi Manyama, May 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A059377, A372966.
Cf. A062952, A373132, A373133.
Cf. A013663, A013667.

f[p_, e_] := (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=4, m=4) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{d|n} J_4(d) * sigma(d^4), where the Jordan totient function J_4(n) = A059377(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(5) * zeta(9) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 - 1/p^8 - 1/p^9 + 1/p^10) = 1.83382546873826519758... . (End)

A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 15, 1, 1, 80, 80, 1, 1, 240, 1280, 240, 1, 1, 624, 9984, 9984, 624, 1, 1, 1200, 49920, 149760, 49920, 1200, 1, 1, 2400, 192000, 1497600, 1497600, 192000, 2400, 1, 1, 3840, 614400, 9216000, 23961600, 9216000, 614400, 3840, 1, 1, 6480, 1658880

Offset: 0

Tom Edgar, Mar 04 2014

We assume that A059383(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_4 given in A059377.
Another name might be the 4-totienomial coefficients.

			The first five terms in the fourth Jordan totient function are 1,15,80,240,624 and so T(4,2) = 240*80*15*1/((15*1)*(15*1))=1280 and T(5,3) = 624*240*80*15*1/((80*15*1)*(15*1))=9984.
The triangle begins
1
1 1
1 15  1
1 80  80   1
1 240 1280 240  1
1 624 9984 9984 624 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A059377, A059383, A238453, A238688, A238743.

q=100 #change q for more rows
P=[0]+[i^4*prod([1-1/p^4 for p in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A059383(n)/(A059383(k)* A059383(n-k)).
T(n,k) = prod_{i=1..n} A059377(i)/(prod_{i=1..k} A059377(i)*prod_{i=1..n-k} A059377(i)).
T(n,k) = A059377(n)/n*(k/A059377(k)*T(n-1,k-1)+(n-k)/A059377(n-k)*T(n-1,k)).

A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

Original entry on oeis.org

0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561, 9375, 14641, 19440, 28561, 36015, 50625, 61440, 83521, 98415, 130321, 150000, 194481, 219615, 279841, 311040, 390625, 428415, 531441, 576240, 707281, 759375, 923521, 983040, 1185921, 1252815, 1500625, 1574640, 1874161, 1954815

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A285989.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000583, A016756, A026741, A059377, A285989, A308422, A309335.

a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 15 x + 76 x^2 + 165 x^3 + 230 x^4 + 165 x^5 + 76 x^6 + 15 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]
Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]

G.f.: x * (1 + 15*x + 76*x^2 + 165*x^3 + 230*x^4 + 165*x^5 + 76*x^6 + 15*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 - x^(2*k)), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 1/2^s).
a(n) = n^4 * (31 - (-1)^n)/32.
a(n) = Sum_{d|n, n/d odd} J_4(d).
Sum_{n>=1} 1/a(n) = 241*Pi^4/21600 = 1.086832913851601267313987...
Multiplicative with a(2^e) = 15*2^(4*e-4), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020

A309338 a(n) = n^4 if n odd, 7*n^4/8 if n even.

Original entry on oeis.org

0, 1, 14, 81, 224, 625, 1134, 2401, 3584, 6561, 8750, 14641, 18144, 28561, 33614, 50625, 57344, 83521, 91854, 130321, 140000, 194481, 204974, 279841, 290304, 390625, 399854, 531441, 537824, 707281, 708750, 923521, 917504, 1185921, 1169294, 1500625, 1469664, 1874161, 1824494

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A284900.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000583, A016756, A059377, A129194, A193356, A284900, A309337.

a[n_] := If[OddQ[n], n^4, 7 n^4/8]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 14 x + 76 x^2 + 154 x^3 + 230 x^4 + 154 x^5 + 76 x^6 + 14 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 14, 81, 224, 625, 1134, 2401, 3584, 6561}, 39]
Table[n^4 (15 - (-1)^n)/16, {n, 0, 38}]

G.f.: x * (1 + 14*x + 76*x^2 + 154*x^3 + 230*x^4 + 154*x^5 + 76*x^6 + 14*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 + x^k), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 2^(1-s)).
a(n) = n^4 * (15 - (-1)^n)/16.
a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_4(d).
Sum_{n>=1} 1/a(n) = 113*Pi^4/10080 = 1.091986834012130496797...
Multiplicative with a(2^e) = 7*2^(4*e-3), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020

A158949 Inverse Moebius transform of A065958.

Original entry on oeis.org

1, 6, 11, 26, 27, 66, 51, 106, 101, 162, 123, 286, 171, 306, 297, 426, 291, 606, 363, 702, 561, 738, 531, 1166, 677, 1026, 911, 1326, 843, 1782, 963, 1706, 1353, 1746, 1377, 2626, 1371, 2178, 1881, 2862, 1683, 3366, 1851, 3198, 2727, 3186, 2211, 4686, 2501

Offset: 1

Vladeta Jovovic, Mar 31 2009

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002117, A013664, A065958.

A158949 := proc(n) add(numtheory[sigma][2](d)^2*numtheory[mobius](n/d),d=numtheory[divisors](n))/n^2 ; end: seq( A158949(n),n=1..80) ; # R. J. Mathar, Apr 02 2009

a[n_] := Sum[2^PrimeNu[n/d] d^2, {d, Divisors[n]}];
Array[a, 80] (* Jean-François Alcover, Nov 20 2020 *)
f[p_, e_] := (p^(2*e)*(p^2 + 1) - 2)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Dec 05 2022 *)

a(n) = sumdiv(n, d, 2^omega(n/d) * d^2); \\ Daniel Suteu, Mar 07 2019

a(n) = (1/n^2)*Sum_{d|n} sigma_2(d)^2*moebius(n/d).
a(n) = Sum_{d|n} 2^omega(n/d) * d^2. - Daniel Suteu, Mar 07 2019
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e)*(p^2+1) - 2)/(p^2-1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)^2/(3*zeta(6)) = 0.473436... . (End)
Dirichlet g.f.: zeta(s)^2*zeta(s-2)/zeta(2*s). - Amiram Eldar, Jan 06 2023
a(n) = Sum_{1 <= j, k <= n} tau(gcd(j, k, n)^2) = Sum_{d divides n} tau(d^2)* J_2(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024
a(n) = Sum_{d divides n} J_4(d)/J_2(d) = Sum_{1 <= i, j, k, l <= n} 1/(J_2(n/gcd(i,j,k,l,n))), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024

Extended by R. J. Mathar, Apr 02 2009

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.

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

Ilya Gutkovskiy, Feb 17 2020

nonn

Wikipedia, Jordan's totient function

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

[&+[&+[MoebiusMu(k div d)*d^n:d in Divisors(k)]:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020

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}]

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

cp's OEIS Frontend

A255819 E.g.f.: exp(Sum_{k>=1} k^3 * x^k).

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A160891 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5.

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A338549 a(n) = n^4 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^4.

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A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).

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A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

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A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

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A309338 a(n) = n^4 if n odd, 7*n^4/8 if n even.

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A158949 Inverse Moebius transform of A065958.

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