A007434 -id:A007434 - cp's OEIS frontend

A046692 Dirichlet inverse of sigma function (A000203).

1, -3, -4, 2, -6, 12, -8, 0, 3, 18, -12, -8, -14, 24, 24, 0, -18, -9, -20, -12, 32, 36, -24, 0, 5, 42, 0, -16, -30, -72, -32, 0, 48, 54, 48, 6, -38, 60, 56, 0, -42, -96, -44, -24, -18, 72, -48, 0, 7, -15, 72, -28, -54, 0, 72, 0, 80, 90, -60, 48, -62, 96, -24, 0, 84, -144, -68, -36, 96, -144, -72, 0, -74, 114, -20, -40, 96, -168

Offset: 1

Andrew R. Feist (arf22540(AT)cmsu2.cmsu.edu)

			a(36) = a(2^2*3^2) = 2*3 = 6.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
Andrew R. Feist, Fun With the Sigma-Function, unpub.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Cf. A000203, A053822, A053825, A053826, A178448.

t := 1; a := proc(n,t) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 then t1 := t1+d^t*mobius(d)*mobius(n/d); fi; od; t1; end;

a[n_] := (k = 0; Do[If[Mod[n, d] == 0, k = k + d*MoebiusMu[d]*MoebiusMu[n/d]], {d, 1, n}]; k); Table[a[n], {n, 1, 78}](* Jean-François Alcover, Oct 13 2011, after Maple *)
f[p_, e_] := Which[e == 1, -p-1, e == 2, p, e >= 3, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=if(n<1,0,direuler(p=2,n,(1-X)*(1-p*X))[n]) /* Ralf Stephan */

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n)))} \\ Andrew Howroyd, Aug 05 2018

a(p) = -p-1, a(p^2) = p, a(p^k) = 0 for k > 2.
Dirichlet g.f.: 1/(zeta(s)*zeta(s-1)). - Benedict W. J. Irwin, Jul 10 2018
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} sigma(k)*A(x^k). - Ilya Gutkovskiy, May 11 2019
From Peter Bala, Jan 17 2024: (Start)
a(n) = Sum_{d divides n} d*mu(d)*mu(n/d). See Brown, p. 408.
a(n) = - Sum_{d divides n, d < n} a(d)*sigma_1(n/d).
a(n) = Sum_{d divides n} d*a(d)*J_2(n/d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{d divides n} d*A007427(d)*phi(n/d), where A007427 is the Dirichlet inverse of the tau function.
More generally, a(n) = Sum_{d divides n} d*sigma_[r]^(-1)(d)*J_(r+1)(n/d), where sigma_[r]^(-1) denotes the Dirichlet inverse of the function sigma_[r] = Sum_{d divides n} d^r.
a(n) = Sum_{k = 1..n} gcd(k, n)*A007427(gcd(k, n)).
a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)*a(gcd(j, k, n)). (End)
Sum_{k=1..n} abs(a(k)) ~ 45*n^2/Pi^4. - Vaclav Kotesovec, May 30 2024

Corrected by T. D. Noe, Nov 13 2006

A059377 Jordan function J_4(n).

Original entry on oeis.org

1, 15, 80, 240, 624, 1200, 2400, 3840, 6480, 9360, 14640, 19200, 28560, 36000, 49920, 61440, 83520, 97200, 130320, 149760, 192000, 219600, 279840, 307200, 390000, 428400, 524880, 576000, 707280, 748800, 923520, 983040, 1171200, 1252800, 1497600, 1555200, 1874160

Offset: 1

N. J. A. Sloane, Jan 28 2001

This sequence is multiplicative. - Mitch Harris, Apr 19 2005

For n = 4 or n >= 6, a(n) is divisible by 240. - Jianing Song, Apr 06 2019

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

T. D. Noe, Table of n, a(n) for n=1..1000
D. H. Lehmer, On a theorem of von Sterneck, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)
Michael Lugo, A little number theory problem (2008)
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
Wikipedia, Jordan's totient function.

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059378 (J_5), A069091 - A069095 (J_6 through J_10).

Cf. A013663.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end:
seq(J(n,4), n=1..40);

JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 4]; Array[f, 38]
f[p_, e_] := p^(4*e) - p^(4*(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^4*moebius(n/d)),","))

a(n)=if(n<1,0,sumdiv(n,d,d^4*moebius(n/d)))

a(n)=if(n<1,0,dirdiv(vector(n,k,k^4),vector(n,k,1))[n])

{ for (n = 1, 1000, write("b059377.txt", n, " ", sumdiv(n, d, d^4*moebius(n/d))); ) } \\ Harry J. Smith, Jun 26 2009

a(n) = Sum_{d|n} d^4*mu(n/d). - Benoit Cloitre, Apr 05 2002
Multiplicative with a(p^e) = p^(4e)-p^(4(e-1)).
Dirichlet generating function: zeta(s-4)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = Sum_{k=1..n} gcd(k,n)^4 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
a(n) = n^4*Product_{distinct primes p dividing n} (1 - 1/p^4). - Tom Edgar, Jan 09 2015
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. - Ilya Gutkovskiy, Apr 25 2017
Sum_{k=1..n} a(k) ~ n^5 / (5*zeta(5)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^4 = 1/zeta(5).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^4/(p^4-1)^2) = 1.0870036174... (End)
O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 11*x^n + 11*x^(2*n) + x^(3*n))/(1 - x^n)^5 = x + 15*x^2 + 80*x^3 + 240*x^4 + 624*x^5 + .... - Peter Bala, Jan 31 2022
From Peter Bala, Jan 01 2024: (Start)
a(n) = Sum_{d divides n} d * J_3(d) * J_1(n/d) = Sum_{d divides n} d^2 * J_2(d) * J_2(n/d) = Sum_{d divides n} d^3 * J_1(d) * J_3(n/d), where J_1(n) = phi(n) = A000010(n), J_2(n) = A007434(n) and J(3,n) = A059376(n).
a(n) = Sum_{k = 1..n} gcd(k, n) * J_3(gcd(k, n)) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_2(gcd(j, k, n)) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n)^3 * J_1(gcd(i, j, k, n)). (End)
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} J_2(i) * J_2(j) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i) * J_3(j) (apply Lehmer, Theorem 1). - Peter Bala, Jan 29 2024

A065465 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

Original entry on oeis.org

8, 8, 1, 5, 1, 3, 8, 3, 9, 7, 2, 5, 1, 7, 0, 7, 7, 6, 9, 2, 8, 3, 9, 1, 8, 2, 2, 9, 0, 3, 2, 2, 7, 8, 4, 7, 1, 2, 9, 8, 6, 9, 2, 5, 7, 2, 0, 8, 0, 7, 6, 7, 3, 3, 6, 7, 0, 1, 6, 8, 5, 3, 5, 5, 4, 8, 6, 5, 7, 9, 0, 6, 3, 7, 9, 4, 1, 6, 9, 7, 4, 1, 0, 2, 2, 0, 4, 5, 5, 1, 7, 9, 7, 0, 2, 0, 9, 6

Offset: 0

N. J. A. Sloane, Nov 19 2001

From Richard R. Forberg, May 22 2023: (Start)
This constant is the asymptotic mean of (phi(n)/n)*(sigma(n)/n), where phi is the Euler totient function (A000010) and sigma is the sum-of-divisors function (A000203).
In contrast, the product of the separate means, mean(phi(n)/n) * mean(sigma(n)/n), converges to 1, with the asymptotic mean(sigma(n)/n) = Pi^2/6 = zeta(2). See A013661.
Also see A062354. (End)

			0.88151383972517077692839182290...

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Steven R. Finch, Class number theory, page 7. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50 and 85.
R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011, Table 5, constant Q_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Simon Plouffe, Generalized expansions of real numbers, 2006.
Eric Weisstein's World of Mathematics, Quadratic Class Number Constant.
Eric Weisstein's World of Mathematics, Prime Products.

Cf. A000010, A007434, A013661, A062354, A078087.

$MaxExtraPrecision = 1000; digits = 98; terms = 1000; LR = Join[{0, 0, 0}, LinearRecurrence[{-2, -1, 1, 1}, {-3, 4, -5, 3}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*PrimeZetaP[n-1]/(n-1), {n, 4, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Mar 14 2021

Sum_{n>=1} phi(n)/(n*J(n)) = (this constant)*A013661 with phi()=A000010() and J() = A007434() [Cohen, Corollary 5.1.1]. - R. J. Mathar, Apr 11 2011

A069097 Moebius transform of A064987, n*sigma(n).

Original entry on oeis.org

1, 5, 11, 22, 29, 55, 55, 92, 105, 145, 131, 242, 181, 275, 319, 376, 305, 525, 379, 638, 605, 655, 551, 1012, 745, 905, 963, 1210, 869, 1595, 991, 1520, 1441, 1525, 1595, 2310, 1405, 1895, 1991, 2668, 1721, 3025, 1891, 2882, 3045, 2755, 2255, 4136, 2737

Offset: 1

Benoit Cloitre, Apr 05 2002

Equals A127569 * [1, 2, 3, ...]. - Gary W. Adamson, Jan 19 2007
Equals row sums of triangle A143309 and of triangle A143312. - Gary W. Adamson, Aug 06 2008
Dirichlet convolution of A000290 and A000010 (see Jovovic formula). - R. J. Mathar, Feb 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wenchang Chu, Jordan's totient function and trigonometric sums, Studia Scientiarum Mathematicarum Hungarica 57 (1), 40-53 (2020).
Nikolai Osipov, Problem 12003, Amer. Math. Monthly, 124(8) (2017), p. 754.

Column 2 of A343510.
Cf. A018804, A033457, A068963, A064987, A127569, A143309, A143312, A360428.
For Sum_{k = 1..n} gcd(k,n)^m see A018804 (m = 1), A343497 (m = 3), A343498 (m = 4) and A343499 (m = 5).

A069097[n_]:=n^2*Plus @@((EulerPhi[#]/#^2)&/@ Divisors[n]); Array[A069097, 100] (* Enrique Pérez Herrero, Feb 25 2012 *)
f[p_, e_] := p^(e-1)*(p^e*(p+1)-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

for(n=1,100,print1((sumdiv(n,k,k*sigma(k)*moebius(n/k))),","))

a(n) = Sum_{d|n} d^2*phi(n/d). - Vladeta Jovovic, Jul 31 2002
a(n) = Sum_{k=1..n} gcd(n, k)^2. - Vladeta Jovovic, Aug 27 2003
Dirichlet g.f.: zeta(s-2)*zeta(s-1)/zeta(s). - R. J. Mathar, Feb 03 2011
a(n) = n*Sum_{d|n} J_2(d)/d, where J_2 is A007434. - Enrique Pérez Herrero, Feb 25 2012.
G.f.: Sum_{n >= 1} phi(n)*(x^n + x^(2*n))/(1 - x^n)^3 = x + 5*x^2 + 11*x^3 + 22*x^4  + .... - Peter Bala, Dec 30 2013
Multiplicative with a(p^e) = p^(e-1)*(p^e*(p+1)-1). - R. J. Mathar, Jun 23 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)). - Vaclav Kotesovec, Sep 18 2020
a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
From Peter Bala, Dec 26 2023: (Start)
For n odd, a(n) = Sum_{k = 1..n} gcd(k,n)/cos(k*Pi/n)^2 (see Osipov and also Chu, p. 51).
It appears that for n odd, Sum_{k = 1..n} (-1)^(k+1)*gcd(k,n)/cos(k*Pi/n)^2 = n. (End)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n). Cf. A360428. - Peter Bala, Jan 16 2024
Sum_{k=1..n} a(k)/k ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 11 2024

A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

Original entry on oeis.org

1, 1, 5, 14, 40, 101, 266, 649, 1593, 3765, 8813, 20168, 45649, 101591, 223654, 486046, 1045541, 2225167, 4692421, 9804734, 20318249, 41766843, 85218989, 172628766, 347338117, 694330731, 1379437080, 2724353422, 5350185097, 10449901555, 20304465729, 39254599832

Offset: 0

Olivier Gérard

In general, if g.f. = Product_{k>=1} 1/(1 - x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(4*Pi * c2^(1/4) * n^(3/4) / (3*15^(1/4)) + c1*Zeta(3) / Pi^2 * sqrt(15*n/c2) + (Pi * 5^(1/4) * c0 / (2*3^(3/4) * c2^(1/4)) - 15^(5/4) * c1^2 * Zeta(3)^2 / (2*c2^(5/4) * Pi^5)) * n^(1/4) + c1/12 + 75 * c1^3 * Zeta(3)^3 / (c2^2 * Pi^8) - 5*c0 * c1 * Zeta(3) / (4*c2 * Pi^2) - c2*Zeta(3) / (4*Pi^2)) * Pi^(c1/12) * (c2/15)^(1/8 + c0/8 + c1/48) / (A^c1 * 2^((c0 + 3)/2) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Let A(x) = Product_{k >= 1} (1 - x^k)^(-k^2). The sequence defined by u(n) := [x^n] A(x)^n is conjectured to satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7 and all positive integers n and r. See A380290. - Peter Bala, Feb 02 2025
a(n) is the number of partitions of n where there are k^2 sorts of part k. - Joerg Arndt, Feb 02 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

Euler transform of squares (A000290).
Cf. A000219, A023872-A023878, A294530, A380290.
Column k=2 of A144048. - Alois P. Heinz, Nov 02 2012

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 29 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^2, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35); # Alois P. Heinz, Nov 02 2012

max = 31; Series[ Product[ 1/(1-x^k)^k^2, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^2)) \\ G. C. Greubel, Oct 29 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^2)
print([b(n) for n in range(32)]) # Peter Luschny, Nov 11 2020

a(n) = 1/n * Sum_{k=1..n} a(n-k)*sigma_3(k), n > 0, a(0)=1, where sigma_3(n) = A001158(n) = sum of cubes of divisors of n. - Vladeta Jovovic, Jan 20 2002
G.f.: Prod_{n>=1} exp(sigma_3(n)*x^n/n), where sigma_3(n) is the sum of cubes of divisors of n (=A001158(n)). - N-E. Fahssi, Mar 28 2010
G.f. (conjectured): 1/Product_{n>=1} E(x^n)^J2(n) where E(x) = Product_{n>=1} 1-x^n and J2(n) = A007434(n) [follows from the identity Sum_{d|n} J2(d) = n^2 - Peter Bala, Feb 02 2025]. - Joerg Arndt, Jan 25 2011
a(n) ~ exp(4 * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) / (4*Pi^2)) / (2^(3/2) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 27 2015

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A082579 Expansion of e.g.f.: exp( x/(1-x)^2 ).

Original entry on oeis.org

1, 1, 5, 31, 241, 2261, 24781, 309835, 4342241, 67308841, 1141960501, 21026890391, 417264626065, 8871853115581, 201100863674621, 4838817223845571, 123128720142540481, 3302478863343928145, 93091427773284348901, 2750635764338982054031, 84994418675445218025521

Offset: 0

Emanuele Munarini, May 07 2003

Old name: A binomial sum.
a(n) is the number of ways that n people can form any number of lines and then designate one person in each line.  Equivalently, number of ways to linearly arrange the elements in each block of a set partition, then underline one element in each block summed over all set partitions of {1,2,...,n}. a(2) = 5: [1'][2'], [1',2], [1,2'], [2',1], [2,1']. - Geoffrey Critzer, Nov 04 2012
It appears that the sequence 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 for each k 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

Seiichi Manyama, Table of n, a(n) for n = 0..433
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000262, A052897, A255806, A255807, A255819.

A082579:= func< n | n eq 0 select 1 else (&+[Factorial(n)*Binomial(n+k-1, n-k)/Factorial(k): k in [1..n]]) >;
[A082579(n): n in [0..25]]; // G. C. Greubel, Feb 23 2021

nn=20;Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)^2],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 04 2012 *)
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)
Table[If[n==0, 1, n*n!*HypergeometricPFQ[{1-n, n+1}, {3/2, 2}, -1/4]], {n, 0, 25}] (* G. C. Greubel, Feb 23 2021 *)

a(n):=n!*sum(binomial(n+k-1,2*k-1)/k!,k,1,n); /* Vladimir Kruchinin, Apr 21 2011 */

my(x='x+O('x^33));
Vec(serlaplace(exp( x/(1-x)^2 )))
/* Joerg Arndt, Sep 14 2012 */

[1 if n==0 else factorial(n)*sum( binomial(n+k-1, n-k)/factorial(k) for k in (1..n)) for n in (0..25)] # G. C. Greubel, Feb 23 2021

a(n) = n!*Sum_{k=0..n} binomial(n+k-1, 2*k-1)/k!.
Recurrence: a(n+3) - (3*n+7)*a(n+2) + (n+2)*(3*n+2)*a(n+1) - (n+2)*(n+1)*n*a(n) = 0.
E.g.f.: exp( x/( 1 - x )^2 ).
Special values of the hypergeometric function 2F2: a(n)=n!*n*hypergeom([n+1, -n+1], [3/2, 2], -1/4), n >= 1. - Karol A. Penson, Jan 29 2004
a(n) ~ 2^(1/6)*n^(n-1/6)*exp(-1/12 + 3*(n/2)^(2/3) - n)/sqrt(3). - Vaclav Kotesovec, Jun 26 2013
E.g.f.: E(0)/2, where E(k) = 1 + 1/( 1 - x/(x + (1-x)^2*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
E.g.f.: exp(Sum_{k>=1} k*x^k). - Vaclav Kotesovec, Mar 07 2015
a(n) = n!*y(n), with y(0) = 1, y(n) = (Sum_{k=0..n-1} (n-k)^2*y(k))/n. - Benedict W. J. Irwin, Jun 02 2016
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_2(k)/k), where J_2() is the Jordan function (A007434). - Ilya Gutkovskiy, May 25 2019
a(n) = n*n!*Hypergeometric2F2([1-n, n+1], [3/2, 2], -1/4) with a(0) = 1. - G. C. Greubel, Feb 23 2021

A000056 Order of the group SL(2,Z_n).

Original entry on oeis.org

1, 6, 24, 48, 120, 144, 336, 384, 648, 720, 1320, 1152, 2184, 2016, 2880, 3072, 4896, 3888, 6840, 5760, 8064, 7920, 12144, 9216, 15000, 13104, 17496, 16128, 24360, 17280, 29760, 24576, 31680, 29376, 40320, 31104, 50616, 41040, 52416, 46080, 68880, 48384, 79464

Offset: 1

N. J. A. Sloane

The number of equivalence classes of matrices modulo n of integer matrices with determinant 1 modulo n. - Michael Somos, Mar 20 2004
24 | a(n) if n > 2. - Michael Somos, Nov 15 2011
A divisibility sequence, that is, a(n) divides a(n*m) for all positive integers n and m. - Michael Somos, Jan 01 2017
The group SL(2,Z_2) is isomorphic to the symmetric group S_3. - Bernard Schott, Mar 15 2020
a(n) = [SL_2(Z) : Gamma(n)], index of the principal congruence subgroup of the special linear group over integers. - Andrey Zabolotskiy, Feb 14 2025

			G.f. = x + 6*x^2 + 24*x^3 + 48*x^4 + 120*x^5 + 144*x^6 + 336*x^7 +384*x^8 + ...
a(2) = 6 because [0, 1; 1, 0], [0, 1; 1, 1], [1, 0; 0, 1], [1, 0; 1, 1], [1, 1; 0, 1], [1, 1; 1, 0] are the six matrices modulo 2 with determinant 1 modulo 2.

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 46.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 75.

T. D. Noe, Table of n, a(n) for n = 1..1000
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Wikipedia, Congruence subgroup.
Index to divisibility sequences
Index entries for sequences related to groups

Cf. A001766.
Row n=2 of A316623.
Row sums of A316564.
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A007434 ([SL_2(Z) : Gamma_1(n)]), A001615 ([SL_2(Z) : Gamma_0(n)]).

proc(n) local b,d: b := n^3: for d from 1 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1-d^(-2)): fi: od: RETURN(b): end:

(* From Olivier Gérard, Aug 15 1997: (Start) *)
Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1-1/#2^2), #1 ]&, n^3, Range[ n ] ], {n, 1, 35} ]
Table[ n^3 Times@@(1-1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]^2), {n, 1, 35} ]  (* End *)
a[ n_] := If[ n<1, 0, n Sum[ d^2 MoebiusMu[ n/d ], {d, Divisors @ n}]]; (* Michael Somos, Nov 15 2011 *)
Table[ n DirichletConvolve[ MoebiusMu[m], m^2, m, n], {n, 1, 35}] (* Li Han, Mar 15 2020 *)
a[n_] := #.RotateLeft[#] & @ Sort[Mod[ Outer[Times, Range[n], Range[n]], n] // Flatten // Tally][[;; , 2]]
Table[a[n], {n, 1, 35}] (* Li Han, Mar 15 2020 *)

{a(n) = if( n<1, 0, n * sumdiv(n, d, d^2 * moebius(n / d)))}; /* Michael Somos, Mar 05 2008 */

from math import prod
from sympy import factorint
def A000056(n): return prod((p+1)*(p-1)*p**(3*e-2) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = (p^2 - 1)*p^(3e-2). - David W. Wilson, Aug 01 2001
a(n) = A000252(n)/phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
a(n) = n*Sum_{d|n} d^2*mu(n/d) = n*A007434(n) where A007434 is the Jordan function J_2(n). - Benoit Cloitre, May 03 2003
a(n) = A007434(n^2)/n. - Enrique Pérez Herrero, Sep 14 2010
a(n) = A007434(n^3)/n^3. - Enrique Pérez Herrero, Dec 19 2010
Dirichlet g.f. zeta(s-3)/zeta(s-1). - R. J. Mathar, Feb 27 2011
A046970(n) divides a(n). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ n^4 / (4*Zeta(3)). - Vaclav Kotesovec, Jan 30 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2 / ((p-1)^2 * (p+1) * (p^2 + p + 1))) = 1.258448350408311046314826069717731136828991478925039589864338603650639811... - Vaclav Kotesovec, Sep 19 2020

More terms from Vaclav Kotesovec, Sep 19 2020

A059378 Jordan function J_5(n).

Original entry on oeis.org

1, 31, 242, 992, 3124, 7502, 16806, 31744, 58806, 96844, 161050, 240064, 371292, 520986, 756008, 1015808, 1419856, 1822986, 2476098, 3099008, 4067052, 4992550, 6436342, 7682048, 9762500, 11510052, 14289858, 16671552, 20511148, 23436248, 28629150, 32505856

Offset: 1

N. J. A. Sloane, Jan 28 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

T. D. Noe, Table of n, a(n) for n=1..1000
D. H. Lehmer, On a theorem of von Sterneck, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)
Wikipedia, Jordan's totient function.

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A069091 - A069095 (J_6 through J_10).

Cf. A013664.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 5)

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 5]; Array[f, 30]
f[p_, e_] := p^(5*e) - p^(5*(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^5*moebius(n/d)),","))

{ for (n = 1, 1000, write("b059378.txt", n, " ", sumdiv(n, d, d^5*moebius(n/d))); ) } \\ Harry J. Smith, Jun 26 2009

from sympy import divisors, mobius
def a(n):
    return sum(d**5 * mobius(n // d) for d in divisors(n))
# Indranil Ghosh, Apr 26 2017

a(n) = Sum_{d|n} d^5*mu(n/d). - Benoit Cloitre, Apr 05 2002
Multiplicative with a(p^e) = p^(5e)-p^(5(e-1)).
Dirichlet generating function: zeta(s-5)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = n^5*Product_{distinct primes p dividing n} (1-1/p^5). - Tom Edgar, Jan 09 2015
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^6. - Ilya Gutkovskiy, Apr 25 2017
Sum_{k=1..n} a(k) ~ 315*n^6 / (2*Pi^6). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^5 = 1/zeta(6).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^5/(p^5-1)^2) = 1.0379908060... (End)
O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 26*x^n + 66*x^(2*n) + 26*x^(3*n) + x^(4*n))/(1 - x^n)^6 = x + 31*x^2 + 242*x^3 + 992*x^4 + 3124*x^5 + .... - Peter Bala, Jan 31 2022
From Peter Bala, Jan 01 2024: (Start)
a(n) = Sum_{d divides n} d * J_4(d) * J_1(n/d) = Sum_{d divides n} d^2 * J_3(d) * J_2(n/d) = Sum_{d divides n} d^3 * J_2(d) * J_3(n/d) = Sum_{d divides n} d^4 * J_1(d) * J_4(n/d), where J_1(n) = phi(n) = A000010(n), J_2(n) = A007434(n), J(3,n) = A059376(n) and J_4(n) = A059377(n).
a(n) = Sum_{k = 1..n} gcd(k, n) * J_4(gcd(k, n)).
a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_3(gcd(j, k, n)). (End)
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} J_2(i) * J_3(j) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i) * J_4(j) (apply Lehmer, Theorem 1). - Peter Bala, Jan 30 2024

A063659 The number of integers m in [1..n] for which gcd(m,n) is not divisible by a square greater than 1.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 8, 10, 11, 9, 13, 14, 15, 12, 17, 16, 19, 15, 21, 22, 23, 18, 24, 26, 24, 21, 29, 30, 31, 24, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 33, 40, 46, 47, 36, 48, 48, 51, 39, 53, 48, 55, 42, 57, 58, 59, 45, 61, 62, 56, 48, 65, 66, 67, 51, 69, 70, 71, 48

Offset: 1

Floor van Lamoen, Jul 24 2001

Equals Möbius transform of A001615. - Gary W. Adamson, May 23 2008

The absolute values of the Dirichlet inverse of A007913. - R. J. Mathar, Dec 22 2010

			For n=12 we find only GCD(4,12), GCD(8,12) and GCD(12,12) divisible by 4, so a(12)=9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A generalized Euler phi-function, Math. Mag. 41 (1968), 276-279; this is function phi_2(n).
E. K. Haviland, An analogue of Euler's phi-function, Duke Math. J. 11 (1944), 869-872.
V. L. Klee, Jr., A generalization of Euler's phi function, Amer. Math. Monthly, 55(6) (1948), 358-359; this is function Phi_2(n).
Paul J. McCarthy, On a certain family of arithmetic functions, Amer. Math. Monthly 65 (1958), 586-590; this is function T_2(n).
Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory 8(1) (2008), #A50; see Example 5 with f(d) = mu(d)^2 (cf. the formulas by _Benoit Cloitre_ below).

Cf. A001615.
Absolute values of the Dirichlet inverse of A007913.
Row 2 of A309287.

A063659 := proc(n)
    local a,ep,p,e;
    a := 1 ;
    for ep in ifactors(n)[2] do
        p := op(1,ep) ;
        e := op(2,ep) ;
        if e = 1 then
            a := a*p ;
        else
            a := a*(p^e-p^(e-2)) ;
        end if;
    end do ;
    a ;
end proc:
seq(A063659(n),n=1..100) ; # R. J. Mathar, Jul 04 2019

nn = 72; f[list_, i_] := list[[i]]; a =Table[If[Max[FactorInteger[n][[All, 2]]] < 2, 1, 0], {n, 1, nn}]; b =Table[EulerPhi[n], {n, 1, nn}]; Table[
DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 22 2015 *)
f[p_, e_] := If[e == 1, p, p^e - p^(e-2)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 29 2020 *)

a(n)=sum(k=1,n,moebius(gcd(n,k))^2) \\ Benoit Cloitre, Jun 14 2007

for (n=1, 2000, a=1; for (m=2, n, if (issquarefree(gcd(m, n)), a++)); write("b063659.txt", n, " ", a) ) \\ Harry J. Smith, Aug 27 2009

a(n)=my(f=factor(n)); prod(i=1,#f~, if(f[i,2]>1, f[i,1]^(f[i,2]-2) * (f[i,1]^2 - 1), f[i,1])) \\ Charles R Greathouse IV, Jan 08 2018

a(n) = n - A063658(n).
Multiplicative with a(p) = p and a(p^e) = p^e-p^(e-2), e>1. - Vladeta Jovovic, Jul 26 2001
a(n) = Sum_{d|n} phi(d)*mu(n/d)^2, Dirichlet convolution of A000010 and A008966. - Benoit Cloitre, Sep 08 2002
a(n) = Sum_{k = 1..n} mu(gcd(n,k))^2. - Benoit Cloitre, Jun 14 2007
Dirichlet g.f.: zeta(s-1)/zeta(2s). - R. J. Mathar, Feb 27 2011
a(n) = Sum_{k=1..n} psi(gcd(k,n)) * cos(2*Pi*k/n), where psi is A001615. - Enrique Pérez Herrero, Jan 18 2013
Sum_{k=1..n} a(k) ~ 45*n^2 / Pi^4. - Vaclav Kotesovec, Jan 11 2019 [This is a special case of a general result by McCarthy (1958), which was reproved later by Cohen (1968). - Petros Hadjicostas, Jul 20 2019]
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
G.f.: Sum_{k>=1} mu(k) * x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 20 2021
a(n) = Sum_{d divides n} mobius(n/d)*J_2(d)/phi(d); that is, the Dirichlet convolution of the Möbius function A008683(n) and the Dedekind psi function A001615(n), and where the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 23 2024

More terms from Vladeta Jovovic and Dean Hickerson, Jul 26 2001

Name edited by Petros Hadjicostas, Jul 21 2019

A023023 Number of partitions of n into 3 unordered relatively prime parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 10, 8, 14, 12, 16, 16, 24, 18, 30, 24, 32, 30, 44, 32, 50, 42, 54, 48, 70, 48, 80, 64, 80, 72, 96, 72, 114, 90, 112, 96, 140, 96, 154, 120, 144, 132, 184, 128, 196, 150, 192, 168, 234, 162, 240, 192, 240, 210, 290, 192, 310, 240, 288, 256, 336, 240, 374

Offset: 3

David W. Wilson

nonn

			From _Gus Wiseman_, Oct 08 2020: (Start)
The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11):
  111   211   221   321   322   332   432   433   443   543   544
              311   411   331   431   441   532   533   552   553
                          421   521   522   541   542   651   643
                          511   611   531   631   551   732   652
                                      621   721   632   741   661
                                      711   811   641   831   733
                                                  722   921   742
                                                  731   A11   751
                                                  821         832
                                                  911         841
                                                              922
                                                              931
                                                              A21
                                                              B11
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 3..10000
Mohamed El Bachraoui, Relatively Prime Partitions with Two and Three Parts, Fibonacci Quart. 46/47 (2008/2009), no. 4, 341-345.

Cf. A023024-A023030, A000742-A000743, A023032-A023035.
A000741 is the ordered version.
A000837 counts these partitions of any length.
A001399(n-3) does not require relative primality.
A023022 is the 2-part version.
A101271 is the strict case.
A284825 counts the case that is also pairwise non-coprime.
A289509 intersected with A014612 gives the Heinz numbers.
A307719 is the pairwise coprime instead of relatively prime version.
A337599 is the pairwise non-coprime instead of relative prime version.
A008284 counts partitions by sum and length.
A078374 counts relatively prime strict partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A007434, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605.

Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&]],{n,3,50}] (* Gus Wiseman, Oct 08 2020 *)

G.f. for the number of partitions of n into m unordered relatively prime parts is Sum(moebius(k)*x^(m*k)/Product(1-x^(i*k), i=1..m), k=1..infinity). - Vladeta Jovovic, Dec 21 2004
a(n) = (n^2/12)*Product_{prime p|n} (1 - 1/p^2) = A007434(n)/12 for n > 3 (proved by Mohamed El Bachraoui). [Jonathan Sondow, May 27 2009]
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,k,n-i-k)). - Wesley Ivan Hurt, Jan 02 2021

cp's OEIS Frontend

A046692 Dirichlet inverse of sigma function (A000203).

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A059377 Jordan function J_4(n).

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A065465 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

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A069097 Moebius transform of A064987, n*sigma(n).

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A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

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A082579 Expansion of e.g.f.: exp( x/(1-x)^2 ).

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