A175836 -id:A175836 - cp's OEIS frontend

A238498 Triangle read by rows: T(n,k) = A175836(n)/(A175836(k)* A175836(n-k)).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 8, 6, 1, 1, 6, 12, 12, 6, 1, 1, 12, 24, 36, 24, 12, 1, 1, 8, 32, 48, 48, 32, 8, 1, 1, 12, 32, 96, 96, 96, 32, 12, 1, 1, 12, 48, 96, 192, 192, 96, 48, 12, 1, 1, 18, 72, 216, 288, 576, 288, 216, 72, 18, 1, 1, 12, 72, 216, 432, 576, 576, 432, 216, 72, 12, 1

Offset: 0

Tom Edgar, Feb 27 2014

We assume that A175836(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Dedekind psi function A001615.
Another name might be the psi-nomial coefficients.

			The first five terms in the Dedekind psi function are 1,3,4,6,6 and so T(4,2) = 6*4*3*1/((3*1)*(3*1))=8 and T(5,3) = 6*6*4*3*1/((4*3*1)*(3*1))=12.
The triangle begins
1
1  1
1  3  1
1  4  4  1
1  6  8  6  1
1  6  12 12 6 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
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. A001615, A175836, A238453.

a238498 n k = a238498_tabl !! n !! k
a238498_row n = a238498_tabl !! n
a238498_tabl = [1] : f [1] a001615_list where
   f xs (z:zs) = (map (div y) $ zipWith (*) ys $ reverse ys) : f ys zs
     where ys = y : xs; y = head xs * z
-- Reinhard Zumkeller, Mar 01 2014

A175836 := proc(n) option remember; local p;
`if`(n<2,1,n*mul(1+1/p,p=factorset(n))*A175836(n-1)) end:
A238498 := (n,k) -> A175836(n)/(A175836(k)*A175836(n-k)):
seq(seq(A238498(n,k),k=0..n),n=0..10); # Peter Luschny, Feb 28 2014

DedekindPsi[n_] := Sum[MoebiusMu[n/d] d^2 , {d, Divisors[n]}]/EulerPhi[n];
(* b = A175836 *) b[n_] := Times @@ DedekindPsi /@ Range[n];
T[n_, k_] := b[n]/(b[k] b[n-k]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jul 02 2019 *)

q=100 #change q for more rows
P=[0]+[i*prod([(1+1/x) for x 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) = A175836(n)/(A175836(k)*A175836(n-k)).
T(n,k) = prod_{i=1..n} A001615(i)/(prod_{i=1..k} A001615(i)*prod_{i=1..n-k} A001615(i)).
T(n,k) = A001615(n)/n*(k/A001615(k)*T(n-1,k-1)+(n-k)/A001615(n-k)*T(n-1,k)).
T(n,k) = A238688(n,k)/A238453(n,k).

A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, 90, 60, 144, 62, 96, 96, 96, 84, 144, 68, 108, 96

Offset: 1

N. J. A. Sloane

Number of primitive sublattices of index n in generic 2-dimensional lattice; also index of Gamma_0(n) in SL_2(Z).
A generic 2-dimensional lattice L =  consists of all vectors of the form mV + nW, (m,n integers). A sublattice S =  has index |ad-bc| and is primitive if gcd(a,b,c,d) = 1. The generic lattice L has precisely a(2) = 3 sublattices of index 2, namely <2V,W>,  and  (which = ) and so on for other indices.
The sublattices of index n are in 1-to-1 correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is the present sequence.
SL_2(Z) = Gamma is the group of all 2 X 2 matrices [a b; c d] where a,b,c,d are integers with ad-bc = 1 and Gamma_0(N) is usually defined as the subgroup of this for which N|c. But conceptually Gamma is best thought of as the group of (positive) automorphisms of a lattice , its typical element taking V -> aV + bW, W -> cV + dW and then Gamma_0(N) can be defined as the subgroup consisting of the automorphisms that fix the sublattice  of index N. - J. H. Conway, May 05 2001
Dedekind proved that if n = k_i*j_i for i in I represents all the ways to write n as a product, and e_i=gcd(k_i,j_i), then a(n)= sum(k_i / (e_i * phi(e_i)), i in I ) [cf. Dickson, History of the Theory of Numbers, Vol. 1, p. 123].
Also a(n)= number of cyclic subgroups of order n in an Abelian group of order n^2 and type (1,1) (Fricke). - Len Smiley, Dec 04 2001
The polynomial degree of the classical modular equation of degree n relating j(z) and j(nz) is psi(n) (Fricke). - Michael Somos, Nov 10 2006; clarified by Katherine E. Stange, Mar 11 2022
The Mobius transform of this sequence is A063659. - Gary W. Adamson, May 23 2008
The inverse Mobius transform of this sequence is A060648. - Vladeta Jovovic, Apr 05 2009
The Dirichlet inverse of this sequence is A008836(n) * A048250(n). - Álvar Ibeas, Mar 18 2015
The Riemann Hypothesis is true if and only if a(n)/n - e^gamma*log(log(n)) < 0 for any n > 30. - Enrique Pérez Herrero, Jul 12 2011
The Riemann Hypothesis is also equivalent to another inequality, see the Sole and Planat link. - Thomas Ordowski, May 28 2017
An infinitary analog of this sequence is the sum of the infinitary divisors of n (see A049417). - Vladimir Shevelev, Apr 01 2014
Problem: are there composite numbers n such that n+1 divides psi(n)? - Thomas Ordowski, May 21 2017
The sum of divisors d of n such that n/d is squarefree. - Amiram Eldar, Jan 11 2019
Psi(n)/n is a new maximum for each primorial (A002110) [proof in link: Patrick Sole and Michel Planat, Proposition 1 page 2]. - Bernard Schott, May 21 2020
From Jianing Song, Nov 05 2022: (Start)
a(n) is the number of subgroups of C_n X C_n that are isomorphic to C_n, where C_n is the cyclic group of order n. Proof: the number of elements of order n in C_n X C_n is A007434(n) (they are the elements of the form (a,b) in C_n X C_n where gcd(a,b,n) = 1), and each subgroup isomorphic to C_n contains phi(n) generators, so the number of such subgroups is A007434(n)/phi(n) = a(n).
The total number of order-n subgroups of C_n X C_n is A000203(n). (End)

			Let L = <V,W> be a 2-dimensional lattice. The 6 primitive sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V+W,2W>, <2V,2W+V>. Compare A000203.
G.f. = x + 3*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 12*x^8 + 12*x^9 + ...

Tom Apostol, Intro. to Analyt. Number Theory, page 71, Problem 11, where this is called phi_1(n).
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 228.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 220.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (1).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
O. Bordelles and B. Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, Representing and counting the subgroups of the group Z_m X Z_n, arXiv:1211.1797 [math.GR], 2012.
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications 14:2 (2015), 73-88.
F. A. Lewis et al., Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
E. Pérez Herrero, Recycling Hardy & Wright, Average Order of Dedekind Psi Function, Psychedelic Geometry Blogspot.
Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, to appear in Journal of Combinatorics and Number Theory, arXiv:1011.1825 [math.NT], 2010-2011.
Eric Weisstein's World of Mathematics, Dedekind Function
Wikipedia, Dedekind psi function
Index entries for "core" sequences
Index entries for sequences related to sublattices

Other sequences that count lattices/sublattices: A000203 (with primitive condition removed), A003050 (hexagonal lattice instead), A003051, A054345, A160889, A160891.
Cf. A002110, A027748, A124010, A019269.
Cf. A301594.
Cf. A063659 (Möbius transform), A082020 (average order), A156303 (Euler transform), A173290 (partial sums), A175836 (partial products), A203444 (range).
Cf. A210523 (record values).
Algebraic combinations with other core sequences: A000082, A033196, A175732, A291784, A344695.
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), this sequence (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).
Cf. A082695 (Dgf at s=3), A339925 (Dgf at s=4).

import Data.Ratio (numerator)
a001615 n = numerator (fromIntegral n * (product $
            map ((+ 1) . recip . fromIntegral) $ a027748_row n))
-- Reinhard Zumkeller, Jun 03 2013, Apr 12 2012

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[MoebiusMu(k)^2*x^k/(1-x^k)^2: k in [1..2*m]]) )); // G. C. Greubel, Nov 23 2018

A001615 := proc(n) n*mul((1+1/i[1]),i=ifactors(n)[2]) end; # Mark van Hoeij, Apr 18 2012

Join[{1}, Table[n Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] (* T. D. Noe, Jun 11 2006 *)
Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 100}] (* Jan Mangaldan, Aug 22 2013 *)
a[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}]; (* Michael Somos, Jan 10 2015 *)
Table[n Product[1 + 1/p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 08 2021 *)
Table[n DivisorSum[n, MoebiusMu[#]^2/# &], {n, 20}] (* Eric W. Weisstein, Mar 09 2025 *)

{a(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n])};

{a(n) = if( n<1, 0, n * sumdiv( n, d, moebius(d)^2 / d))}; /* Michael Somos, Nov 10 2006 */

a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^f[i,2] + f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Aug 22 2013

a(n) = n * sumdivmult(n, d, issquarefree(d)/d) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import primefactors
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist) # Chai Wah Wu, Jun 03 2021

def A001615(n) : return n*mul(1+1/p for p in prime_divisors(n))
[A001615(n) for n in (1..69)] # Peter Luschny, Jun 10 2012

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s). - Michael Somos, May 19 2000
Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson, Aug 01 2001
a(n) = A003557(n)*A048250(n) = n*A000203(A007947(n))/A007947(n). - Labos Elemer, Dec 04 2001
a(n) = n*Sum_{d|n} mu(d)^2/d, Dirichlet convolution of A008966 and A000027. - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(n/d)^2 * d. - Joerg Arndt, Jul 06 2011
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_2(n)/J_1(n) = J_2(n)/phi(n) = A007434(n)/A000010(n), where J_k is the k-th Jordan Totient Function.
a(n) = (1/phi(n))*Sum_{d|n} mu(n/d)*d^(b-1), for b=3. (End)
a(n) = n / Sum_{d|n} mu(d)/a(d). - Enrique Pérez Herrero, Jun 06 2012
a(n^k)= n^(k-1) * a(n). - Enrique Pérez Herrero, Jan 05 2013
If n is squarefree, then a(n) = A049417(n) = A000203(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{d^2 | n} mu(d) * A000203(n/d^2). - Álvar Ibeas, Dec 20 2014
The average order of a(n) is 15*n/Pi^2. - Enrique Pérez Herrero, Jan 14 2012. See Apostol. - N. J. A. Sloane, Sep 04 2017
G.f.: Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 25 2018
a(n) = Sum_{d|n} 2^omega(d) * phi(n/d), Dirichlet convolution of A034444 and A000010. - Daniel Suteu, Mar 09 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} 2^omega(gcd(n,k)).
a(n) = Sum_{k=1..n} 2^omega(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = abs(A158523(n)) = A158523(n) * A008836(n). - Enrique Pérez Herrero, Nov 07 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^2). - Ridouane Oudra, Mar 26 2025

More terms from Olivier Gérard, Aug 15 1997

A173290 Partial sums of A001615.

Original entry on oeis.org

1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, 1494, 1566, 1638, 1686, 1782, 1838, 1928, 2000, 2084

Offset: 1

Jonathan Vos Post, Feb 15 2010

nonn

a(n) is even for n >= 2. - Jianing Song, Nov 24 2018

W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88; http://scientificadvances.co.in; DOI: http://dx.doi.org/10.18642/jantaa_7100121599

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88.

Cf. A001615, A063659.
Cf. A082020.
Cf. A175836 (partial products of the Dedekind psi function).

[(&+[MoebiusMu(k)^2*Floor(n/k)*Floor(1 + n/k): k in [1..n]])/2: n in [1..60]]; // G. C. Greubel, Nov 23 2018

with(numtheory): a:=n->(1/2)*add(mobius(k)^2*floor(n/k)*floor(1+n/k),k=1..n); seq(a(n),n=1..55); # Muniru A Asiru, Nov 24 2018

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k], {k,1,n}], {n,60}] (* G. C. Greubel, Nov 23 2018 *)
psi[n_] := If[n==1, 1, n*Times@@(1 + 1/FactorInteger[n][[;;,1]])]; Accumulate[Array[psi, 50]] (* Amiram Eldar, Nov 23 2018 *)

S(n) = sum(k=1, sqrtint(n), moebius(k)*(n\(k*k))); \\ see: A013928
a(n) = sum(k=1, sqrtint(n), k*(k+1) * (S(n\k) - S(n\(k+1))))/2 + sum(k=1, n\(1+sqrtint(n)), moebius(k)^2*(n\k)*(1+n\k))/2; \\ Daniel Suteu, Nov 23 2018

def A173290(n) :
    return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n))
[A173290(n) for n in (1..52)]  # Peter Luschny, Jun 10 2012

a(n) = Sum_{i=1..n} A001615(i) = Sum_{i=1..n} (n * Product_{p|n, p prime} (1 + 1/p)).
a(n) = 15*n^2/(2*Pi^2) + O(n*log(n)). - Enrique Pérez Herrero, Jan 14 2012
a(n) = Sum_{i=1..n} A063659(i) * floor(n/i). - Enrique Pérez Herrero, Feb 23 2013
a(n) = (1/2)*Sum_{k=1..n} mu(k)^2 * floor(n/k) * floor(1+n/k), where mu(k) is the Moebius function. - Daniel Suteu, Nov 19 2018
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (A013928(1+floor(n/k)) - A013928(1+floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k)^2 * floor(n/k) * floor(1+n/k))/2. - Daniel Suteu, Nov 23 2018

A059381 Product J_2(i), i=1..n.

Original entry on oeis.org

1, 3, 24, 288, 6912, 165888, 7962624, 382205952, 27518828544, 1981355655168, 237762678620160, 22825217147535360, 3834636480785940480, 552187653233175429120, 106020029420769682391040, 20355845648787779019079680, 5862483546850880357494947840

Offset: 1

N. J. A. Sloane, Jan 28 2001

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j)^2 for 1 <= i,j <= n. - Avi Peretz, (njk(AT)netvision.net.il), Mar 22 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 203, #17.

G. C. Greubel, Table of n, a(n) for n = 1..250
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.

Cf. A001088, A007434, A175836.

f:= n-> LinearAlgebra:-Determinant(Matrix(n,n,(i,j) -> igcd(i,j)^2)):
map(f, [$1..40]); # Robert Israel, Dec 01 2017

JordanTotient[n_,k_:1] := DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n]; A059381[n_]:=Times@@(JordanTotient[#,2]&/@Range[n] ); (* Enrique Pérez Herrero, Dec 29 2010 *)

a(n) = A001088(n)*A175836(n). - Enrique Pérez Herrero, Oct 08 2011

A059382 Product J_3(i), i=1..n.

Original entry on oeis.org

1, 7, 182, 10192, 1263808, 230013056, 78664465152, 35241680388096, 24739659632443392, 21474024560960864256, 28560452666077949460480, 41584019081809494414458880, 91318505903653649734151700480, 218616503133346837463559170949120

Offset: 1

N. J. A. Sloane, Jan 28 2001

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j)^3 for 1 <= i,j <= n. - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 203, #17.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..100
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Eric Weisstein's World of Mathematics, Le Paige's Theorem

Cf. A001088, A059376, A059381, A059383, A059384, A175836.

JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n]; A059382[n_]:=Times@@(JordanTotient[#, 3]&/@Range[n]); (* Enrique Pérez Herrero, Aug 06 2011 *)

A059383 Product J_4(i), i=1..n.

Original entry on oeis.org

1, 15, 1200, 288000, 179712000, 215654400000, 517570560000000, 1987470950400000000, 12878811758592000000000, 120545678060421120000000000, 1764788726804565196800000000000, 33883943554647651778560000000000000, 967725427920736934795673600000000000000

Offset: 1

N. J. A. Sloane, Jan 28 2001

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j)^4 for 1 <= i,j <= n. - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 203, #17.

G. C. Greubel, Table of n, a(n) for n = 1..140
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Eric Weisstein's World of Mathematics, Le Paige's Theorem

Cf. A001088, A059377, A059381, A059382, A059383, A059384, A175836.

JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n]; A059383[n_]:=Times@@(JordanTotient[#, 4]&/@Range[n]); (* Enrique Pérez Herrero, Aug 12 2011 *)

A059384 a(n) = Product_{i=1..n} J_5(i).

Original entry on oeis.org

1, 31, 7502, 7441984, 23248758016, 174412182636032, 2931171141381153792, 93047096712003345973248, 5471727569246068763302821888, 529903984716066283313298482921472, 85341036738522474927606720674503065600, 20487310643596659421020979792003903940198400

Offset: 1

N. J. A. Sloane, Jan 28 2001

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j)^5 for 1 <= i,j <= n. - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 203, #17.

G. C. Greubel, Table of n, a(n) for n = 1..120
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Eric Weisstein's World of Mathematics, Le Paige's Theorem

Cf. A001088, A059378, A059381, A059382, A059383, A175836.

JordanTotient[n_Integer, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &];  f[n_] := Times @@ (JordanTotient[#, 5] & /@ Range[n]); (* Enrique Pérez Herrero *)  Array[f, 11] (* Robert G. Wilson v, Oct 08 2011 *)

A203444 Numbers in range of Dedekind Psi function: A001615.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 36, 38, 42, 44, 48, 54, 56, 60, 62, 68, 72, 74, 80, 84, 90, 96, 98, 102, 104, 108, 110, 112, 114, 120, 126, 128, 132, 138, 140, 144, 150, 152, 158, 160, 162, 164, 168, 174, 176, 180, 182, 186, 192, 194, 198, 200

Offset: 1

Enrique Pérez Herrero, Jan 02 2012

nonn

a(n) is even for n>2

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, J. Comb. Number Theory 3 (2011), no. 1, 33-38, arXiv:1011.1825v2.

Cf. A001615, A175836, A060648.
Cf. A007617.
Cf. A002202.

terms = 100; Clear[seq]; seq[k_] := seq[k] = Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 1, k terms}] // Union // PadRight[#, terms]&;
seq[k = 1]; seq[k++]; While[Print[k]; seq[k] != seq[k-1], k++];
seq[k] (* Jean-François Alcover, Dec 14 2018, after Jan Mangaldan in A001615 *)

A239672 Product_{i=1..n} J_6(i) where J_6(i) = A069091(i).

Original entry on oeis.org

1, 63, 45864, 184923648, 2889247076352, 132512427909808128, 15589822118733106642944, 4022922418094840702998413312, 2135013202351949099169693925638144, 2101519115233451721701919767332732796928, 3722967203782973732098252983015976113725767680

Offset: 1

Tom Edgar, Mar 23 2014

nonn

This is the generalized factorial for A069091.

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j)^6 for 1 <= i,j <= n.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 203, #17.

Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Eric Weisstein's World of Mathematics, Le Paige's Theorem

Cf. A069091, A175836, A059381, A059382, A059383.

q=15 # change q for more terms
J6=[i^6*prod([1-1/p^6 for p in prime_divisors(i)]) for i in [1..q]]
[prod(J6[0:i+1]) for i in [0..q-1]]

cp's OEIS Frontend

A238498 Triangle read by rows: T(n,k) = A175836(n)/(A175836(k)* A175836(n-k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Sage

Formula

A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

Python

Sage

Formula

Extensions

A173290 Partial sums of A001615.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A059381 Product J_2(i), i=1..n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A059382 Product J_3(i), i=1..n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A059383 Product J_4(i), i=1..n.

Views

Author

Keywords

Comments

References