A079612 Largest number m such that a^n == 1 (mod m) whenever a is coprime to m.
2, 24, 2, 240, 2, 504, 2, 480, 2, 264, 2, 65520, 2, 24, 2, 16320, 2, 28728, 2, 13200, 2, 552, 2, 131040, 2, 24, 2, 6960, 2, 171864, 2, 32640, 2, 24, 2, 138181680, 2, 24, 2, 1082400, 2, 151704, 2, 5520, 2, 1128, 2, 4455360, 2, 264, 2, 12720, 2, 86184, 2, 13920
Offset: 1
Keywords
A069093 Jordan function J_8(n).
1, 255, 6560, 65280, 390624, 1672800, 5764800, 16711680, 43040160, 99609120, 214358880, 428236800, 815730720, 1470024000, 2562493440, 4278190080, 6975757440, 10975240800, 16983563040, 25499934720, 37817088000
Offset: 1
Comments
a(n) is divisible by 480 = (2^5)*3*5 = A006863(4), except for n = 1, 2, 3 and 5. See Lugo. - Peter Bala, Jan 13 2024
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- Michael Lugo, A little number theory problem (2008)
- Wikipedia, Jordan's totient function.
Crossrefs
Programs
-
Maple
with(numtheory): seq(add(d^8 * mobius(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 13 2024
-
Mathematica
JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 8]; Array[f, 25] f[p_, e_] := p^(8*e) - p^(8*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)
-
PARI
for(n=1,100,print1(sumdiv(n,d,d^8*moebius(n/d)),","))
Formula
a(n) = Sum_{d|n} d^8*mu(n/d).
Multiplicative with a(p^e) = p^(8e)-p^(8(e-1)).
Dirichlet generating function: zeta(s-8)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^8*Product_{distinct primes p dividing n} (1-1/p^8). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^9 / (9*zeta(9)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^8 = 1/zeta(9).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^8/(p^8-1)^2) = 1.0040927606... (End)
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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..350
Programs
-
Maple
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
-
Mathematica
JordanTotient[n_,k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n]; A067858[n_]:=JordanTotient[n,n]; Array[A067858,20]
Formula
J_n(n) = sum{k|n} mu(n/k) k^n, where mu() is the Moebius function.
A069092 Jordan function J_7(n).
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
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
Links
- Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
- Wikipedia, Jordan's totient function.
Crossrefs
Programs
-
Mathematica
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 *)
-
PARI
for(n=1, 100, print1(sumdiv(n, d, d^7*moebius(n/d)), ", "))
Formula
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).
1, 511, 19682, 261632, 1953124, 10057502, 40353606, 133955584, 387400806, 998046364, 2357947690, 5149441024, 10604499372, 20620692666, 38441386568, 68585259008, 118587876496, 197961811866, 322687697778, 510999738368
Offset: 1
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
- Wikipedia, Jordan's totient function.
Crossrefs
Programs
-
Mathematica
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 *)
-
PARI
for(n=1,100,print1(sumdiv(n,d,d^9*moebius(n/d)),","))
Formula
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)
A115000 a(n) = J_2(n) / 24.
1, 1, 2, 2, 3, 3, 5, 4, 7, 6, 8, 8, 12, 9, 15, 12, 16, 15, 22, 16, 25, 21, 27, 24, 35, 24, 40, 32, 40, 36, 48, 36, 57, 45, 56, 48, 70, 48, 77, 60, 72, 66, 92, 64, 98, 75, 96, 84, 117, 81, 120, 96, 120, 105, 145, 96, 155, 120, 144, 128, 168, 120, 187, 144, 176, 144, 210, 144
Offset: 5
Keywords
Comments
The Jordan function J_m(n) can be defined as multiplicative with J_m(p^e) = (p^m-1)*p^(m*(e-1)). Cf. A059379.
Looking at the sequences J_m(n) for fixed m, one is struck by the fact that all but a few early terms have a common factor, given in A079612. I will refer to this sequence as K(n), following the notation in the paper by Vaughan and Wooley. (The alternate lambda^*(n) in the comment for A006863 is too awkward.)
In fact, K(m) not only divides J_m(n) for all but finitely many n; it also divides Sum_{k=1..n} J_m(k) for all but finitely many n.
The weight of the n-th elliptic division polynomial -- the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, let e1 = b1, e2 = b2*b1, e3 = b3*b1, e4 = b2*b4*b1, e5 = (b2^4*b4 - b3^3)*b1 = b5*e1 and so on be an elliptic divisibility sequence. Let c2 = b2^4*b4, c3 = b3^3, c4 = b4^2 and cn = bn for n>4. Then c5 = c2 - c3, c6 = c5 - c4, c7 = c6*c3 - c5*c4 and so on. Let the weight of c2, c3, c4 each be 1 and weight of a product is sum of the weights of the factors. The weight of cn is a(n) for n>4. - Michael Somos, Aug 12 2008
Examples
G.f.: x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 5*x^11 + 4*x^12 + 7*x^13 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 5..5000
- Index to divisibility sequences
Crossrefs
Cf. A007434.
Programs
-
Magma
function a(n) return n lt 5 select 0 else Dimension( ModularForms( Gamma1(n), 2)) - Dimension( ModularForms( Gamma1(n), 1)); end function; /* Michael Somos, Aug 05 2014 */
-
Mathematica
a[n_] := DivisorSum[n, #^2*MoebiusMu[n/#]&]/24; Table[a[n], {n, 5, 80}] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *) -
PARI
{a(n) = if( n<5, 0, sumdiv(n, d, d^2 * moebius(n / d)) / 24)}; /* Michael Somos, Aug 12 2008 */
Formula
A007434(n) = 24 * a(n) unless n<5. - Michael Somos, Aug 12 2008
Extensions
More terms from Michael Somos, Aug 12 2008
A059409 a(n) = 4^n * (2^n - 1).
0, 4, 48, 448, 3840, 31744, 258048, 2080768, 16711680, 133955584, 1072693248, 8585740288, 68702699520, 549688705024, 4397778075648, 35183298347008, 281470681743360, 2251782633816064, 18014329790005248, 144114913197948928, 1152920405095219200
Offset: 0
Comments
Examples
(4,48,448,3840,...) = (8,64,512,4096,...) - (2,12,56,240,...) - (1,3,7,15,...) - (1,1,1,1,...)
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
Links
- Harry J. Smith, Table of n, a(n) for n = 0..100
- Index entries for linear recurrences with constant coefficients, signature (12,-32).
Programs
-
GAP
List([0..100], n->4^n * (2^n - 1)); # Muniru A Asiru, Jan 29 2018
-
Magma
[4^n*(2^n - 1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011
-
Maple
seq(4^n * (2^n - 1), n=0..20); # Muniru A Asiru, Jan 29 2018
-
Mathematica
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 *) -
PARI
a(n) = { 4^n*(2^n - 1) } \\ Harry J. Smith, Jun 26 2009
Formula
Equals J_n(8) (see A059379).
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
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).
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
Examples
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, ...
Links
- Wikipedia, Dedekind psi function
Crossrefs
Programs
-
Mathematica
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
Formula
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.
1, 1, 4, 34, 456, 12388, 677244, 69513187, 13727785600, 5551190294478, 4378921597198116, 6705804947252051188, 21038823519531799964724, 131183284379709847290156854, 1603688086811508900855649976528, 40293997364837932973226463649637881, 2031337795407293560044987268598542021504
Offset: 0
Keywords
Links
- Wikipedia, Jordan's totient function
Programs
-
Mathematica
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}]
Formula
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.
1, 1, 3, 33, 425, 12083, 665707, 68834806, 13654633905, 5535319947544, 4371956013518511, 6700051541666225780, 21029477920140943174285, 131152064162504305814647983, 1603485136950993248524876767297, 40291404321882574322412345562762188, 2031269423141309839019651314585293713041
Offset: 0
Keywords
Links
- Wikipedia, Jordan's totient function
Programs
-
Mathematica
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}]
Formula
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).
Comments
References
Links
Crossrefs
Programs
PARI
a(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)););); res;);} \\ Michel Marcus, May 12 2018Formula
Extensions