A059380 -id:A059380 - cp's OEIS frontend

A067858 J_n(n), where J is the Jordan function, J_n(n) = n^n product{p|n}(1 - 1/p^n), the product is over the distinct primes, p, dividing n.

1, 3, 26, 240, 3124, 45864, 823542, 16711680, 387400806, 9990233352, 285311670610, 8913906892800, 302875106592252, 11111328602468784, 437893859848932344, 18446462598732840960, 827240261886336764176, 39346257879101671328376, 1978419655660313589123978

Offset: 1

Leroy Quet, Feb 15 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..350

Main diagonal of A059379, A059380.

with(numtheory):
a:= n-> n^n*mul(1-1/p^n, p=factorset(n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 09 2015

JordanTotient[n_,k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n]; A067858[n_]:=JordanTotient[n,n]; Array[A067858,20]

J_n(n) = sum{k|n} mu(n/k) k^n, where mu() is the Moebius function.

A069092 Jordan function J_7(n).

Original entry on oeis.org

1, 127, 2186, 16256, 78124, 277622, 823542, 2080768, 4780782, 9921748, 19487170, 35535616, 62748516, 104589834, 170779064, 266338304, 410338672, 607159314, 893871738, 1269983744, 1800262812, 2474870590, 3404825446, 4548558848

Offset: 1

Benoit Cloitre, Apr 05 2002

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A069091 (J_6), A069092 (J_7), A069093 (J_8), A069094 (J_9), A069095 (J_10). [Enrique Pérez Herrero, Nov 02 2010]
Cf. A013666.

JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
A069092[n_] := JordanTotient[n, 7]; (* Enrique Pérez Herrero, Nov 02 2010 *)
f[p_, e_] := p^(7*e) - p^(7*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1, 100, print1(sumdiv(n, d, d^7*moebius(n/d)), ", "))

a(n) = Sum_{d|n} d^7*mu(n/d).
Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).
Dirichlet generating function: zeta(s-7)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A_7(x^n)/(1 - x^n)^8 = x + 127*x^2 + 2186*x^3 + 16256*x^4 + 78124*x^5 + ..., where A_7(x) = x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7 is the 7th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A069094 Jordan function J_9(n).

Original entry on oeis.org

1, 511, 19682, 261632, 1953124, 10057502, 40353606, 133955584, 387400806, 998046364, 2357947690, 5149441024, 10604499372, 20620692666, 38441386568, 68585259008, 118587876496, 197961811866, 322687697778, 510999738368

Offset: 1

Benoit Cloitre, Apr 05 2002

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).

Cf. A013668.

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 9]; Array[f, 22]
f[p_, e_] := p^(9*e) - p^(9*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(sumdiv(n,d,d^9*moebius(n/d)),","))

a(n) = Sum_{d|n} d^9*mu(n/d).
Multiplicative with a(p^e) = p^(9e)-p^(9(e-1)).
Dirichlet generating function: zeta(s-9)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^9*Product_{distinct primes p dividing n} (1-1/p^9). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 18711*n^10 / (2*Pi^10). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^9 = 1/zeta(10).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^9/(p^9-1)^2) = 1.0020122252...  (End)

A059409 a(n) = 4^n * (2^n - 1).

Original entry on oeis.org

0, 4, 48, 448, 3840, 31744, 258048, 2080768, 16711680, 133955584, 1072693248, 8585740288, 68702699520, 549688705024, 4397778075648, 35183298347008, 281470681743360, 2251782633816064, 18014329790005248, 144114913197948928, 1152920405095219200

Offset: 0

N. J. A. Sloane, Alford Arnold, Jan 30 2001

Jordan's totient functions are described more fully in A059379 and A059380; for example, J_1(n) is Euler's totient function and J_2(n) the Moebius transform of squares.

			(4,48,448,3840,...) = (8,64,512,4096,...) - (2,12,56,240,...) - (1,3,7,15,...) - (1,1,1,1,...)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Cf. A059379, A059380, A016152.
Cf. A024023, A000225, A000012.
Cf. A065442.

List([0..100], n->4^n * (2^n - 1)); # Muniru A Asiru, Jan 29 2018

[4^n*(2^n - 1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

seq(4^n * (2^n - 1), n=0..20); # Muniru A Asiru, Jan 29 2018

Table[4^n*(2^n - 1), {n,0,30}] (* G. C. Greubel, Jan 29 2018 *)
LinearRecurrence[{12,-32},{0,4},20] (* Harvey P. Dale, Oct 14 2019 *)

a(n) = { 4^n*(2^n - 1) } \\ Harry J. Smith, Jun 26 2009

Equals J_n(8) (see A059379).
J_n(8) = 8^n - A024023(n) - A000225(n) - A000012(n).
a(n) = 4*A016152(n).
G.f.: 4*x / ( (8*x-1)*(4*x-1) ). - R. J. Mathar, Nov 23 2018
Sum_{n>0} 1/a(n) = E - 4/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022
a(n) = A291779(A008585(n)) = A045991(A000079(n)). - Mathew Englander, Feb 08 2024

A059410 J_n(9) (see A059379).

Original entry on oeis.org

0, 6, 72, 702, 6480, 58806, 530712, 4780782, 43040160, 387400806, 3486725352, 31380882462, 282429005040, 2541864234006, 22876787671992, 205891117745742, 1853020145805120, 16677181570526406, 150094634909578632, 1350851716510730622, 12157665455570144400, 109418989121052006006

Offset: 0

N. J. A. Sloane, Jan 30 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A000244, A016142, A024023, A059379, A059380, A059409.

[9^n-3^n: n in [0..20]]; // Vincenzo Librandi, Jun 03 2011

A059410:=n->9^n-3^n: seq(A059410(n), n=0..30); # Wesley Ivan Hurt, Aug 16 2016

Table[9^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 3 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)

a(n) = 9^n - 3^n; a(n) = 12*a(n-1) - 27*a(n-2) for n > 1. - Vincenzo Librandi, Jun 03 2011
From Vincenzo Librandi, Oct 04 2014: (Start)
a(n) = 3^n*(3^n-1) = A000244(n)*A024023(n).
G.f.: 6*x/((1-3*x)*(1-9*x)). (End)
a(n) = 6*A016142(n). - R. J. Mathar, Nov 23 2018
E.g.f.: 2*exp(6*x)*sinh(3*x). - Elmo R. Oliveira, Mar 31 2025

A320973 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 9, 10, 6, 2, 1, 17, 28, 20, 6, 4, 1, 33, 82, 72, 26, 12, 2, 1, 65, 244, 272, 126, 50, 8, 2, 1, 129, 730, 1056, 626, 252, 50, 12, 2, 1, 257, 2188, 4160, 3126, 1394, 344, 80, 12, 4, 1, 513, 6562, 16512, 15626, 8052, 2402, 576, 90, 18, 2

Offset: 1

Ilya Gutkovskiy, Oct 25 2018

			Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  2,   3,   5,    9,    17,    33,  ...
  2,   4,  10,   28,    82,   244,  ...
  2,   6,  20,   72,   272,  1056,  ...
  2,   6,  26,  126,   626,  3126,  ...
  4,  12,  50,  252,  1394,  8052,  ...

Wikipedia, Dedekind psi function

Columns k=0..4 give A034444, A001615, A065958, A065959, A065960.

Cf. A008683, A059379, A059380, A320974 (diagonal).

Table[Function[k, n^k Product[1 + Boole[PrimeQ[d]]/d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j]^2 PolyLog[-k, x^j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[n/d]^2 d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} mu(j)^2*PolyLog(-k,x^j), where PolyLog() is the polylogarithm function.

A(n,k) = Sum_{d|n} mu(n/d)^2*d^k.

A321264 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^J_n(k), where J_() is the Jordan function.

Original entry on oeis.org

1, 1, 4, 34, 456, 12388, 677244, 69513187, 13727785600, 5551190294478, 4378921597198116, 6705804947252051188, 21038823519531799964724, 131183284379709847290156854, 1603688086811508900855649976528, 40293997364837932973226463649637881, 2031337795407293560044987268598542021504

Offset: 0

Ilya Gutkovskiy, Nov 01 2018

nonn

Wikipedia, Jordan's totient function

Cf. A059379, A059380, A061255, A301875, A319647, A321265.

Table[SeriesCoefficient[Product[1/(1 - x^k)^Sum[d^n MoebiusMu[k/d], {d, Divisors[k]}], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[Sum[Sum[d j^n MoebiusMu[d/j], {j, Divisors[d]}], {d, Divisors[k]}] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

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

A321265 a(n) = [x^n] Product_{k>=1} (1 + x^k)^J_n(k), where J_() is the Jordan function.

Original entry on oeis.org

1, 1, 3, 33, 425, 12083, 665707, 68834806, 13654633905, 5535319947544, 4371956013518511, 6700051541666225780, 21029477920140943174285, 131152064162504305814647983, 1603485136950993248524876767297, 40291404321882574322412345562762188, 2031269423141309839019651314585293713041

Offset: 0

Ilya Gutkovskiy, Nov 01 2018

nonn

Wikipedia, Jordan's totient function

Cf. A059379, A059380, A299069, A301876, A321042, A321264.

Table[SeriesCoefficient[Product[(1 + x^k)^Sum[d^n MoebiusMu[k/d], {d, Divisors[k]}], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[Sum[Sum[(-1)^(k/d + 1) d j^n MoebiusMu[d/j], {j, Divisors[d]}], {d, Divisors[k]}] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

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

A322324 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Product_{p|n, p prime} (1 - p^k).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -3, -2, 0, 1, -7, -8, -1, 0, 1, -15, -26, -3, -4, 0, 1, -31, -80, -7, -24, 2, 0, 1, -63, -242, -15, -124, 24, -6, 0, 1, -127, -728, -31, -624, 182, -48, -1, 0, 1, -255, -2186, -63, -3124, 1200, -342, -3, -2, 0, 1, -511, -6560, -127, -15624, 7502, -2400, -7, -8, 4, 0

Offset: 1

Ilya Gutkovskiy, Dec 03 2018

			Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -2,  -8,  -26,   -80,  -242, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -4, -24, -124,  -624, -3124, ...
  0,  2,  24,  182,  1200,  7502, ...

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

Columns k=0..5 give A063524, A023900, A046970, A063453, A189922, A189923.

Cf. A008683, A059379, A059380, A321222 (diagonal).

Table[Function[k, Product[1 - Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j] j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[d] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

T(n, k) = sumdiv(n, d, moebius(d)*d^k);
matrix(6, 6, n, k, T(n, k-1)) \\ Michel Marcus, Dec 03 2018

G.f. of column k: Sum_{j>=1} mu(j)*j^k*x^j/(1 - x^j).
Dirichlet g.f. of column k: zeta(s)/zeta(s-k).
A(n,k) = Sum_{d|n} mu(d)*d^k.

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

A067858 J_n(n), where J is the Jordan function, J_n(n) = n^n product{p|n}(1 - 1/p^n), the product is over the distinct primes, p, dividing n.

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A069092 Jordan function J_7(n).

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A069094 Jordan function J_9(n).

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A059409 a(n) = 4^n * (2^n - 1).

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A059410 J_n(9) (see A059379).

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A320973 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).

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A321264 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^J_n(k), where J_() is the Jordan function.

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