A059381 -id:A059381 - cp's OEIS frontend

A238688 Triangle read by rows: T(n,k) = A059381(n)/(A059381(k)*A059381(n-k)).

1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 12, 32, 12, 1, 1, 24, 96, 96, 24, 1, 1, 24, 192, 288, 192, 24, 1, 1, 48, 384, 1152, 1152, 384, 48, 1, 1, 48, 768, 2304, 4608, 2304, 768, 48, 1, 1, 72, 1152, 6912, 13824, 13824, 6912, 1152, 72, 1, 1, 72, 1728, 10368, 41472, 41472

Offset: 0

Tom Edgar, Mar 02 2014

We assume that A059381(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_2 given in A007434.
Another name might be the 2-totienomial coefficients.

			The first five terms in the second Jordan totient function are 1,3,8,12,24 and so T(4,2) = 12*8*3*1/((3*1)*(3*1))=32 and T(5,3) = 24*12*8*3*1/((8*3*1)*(3*1))=96.
The triangle begins
1
1   1
1   3   1
1   8   8   1
1   12  32  12   1
1   24  96  96   24   1
1   24  192 288  192  24 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. A007434, A059381, A238453.

q=100 #change q for more rows
P=[0]+[i^2*prod([1-1/p^2 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) = A059381(n)/(A059381(k)* A059381(n-k)).
T(n,k) = prod_{i=1..n} A007434(i)/(prod_{i=1..k} A007434(i)*prod_{i=1..n-k} A007434(i)).
T(n,k) = A007434(n)/n*(k/A007434(k)*T(n-1,k-1)+(n-k)/A007434(n-k)*T(n-1,k)).

A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

Original entry on oeis.org

1, 1, 1, 2, 4, 16, 32, 192, 768, 4608, 18432, 184320, 737280, 8847360, 53084160, 424673280, 3397386240, 54358179840, 326149079040, 5870683422720, 46965467381760, 563585608581120, 5635856085811200, 123988833887846400, 991910671102771200, 19838213422055424000

Offset: 0

Simon Plouffe

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j) for 1 <= i,j <= n [Smith and Mansion]. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001

The matrix M(i,j) = gcd(i,j) is sequence A003989. - Michael Somos, Jun 25 2012

			a(2) = 1 because the matrix M is: [1,1; 1,2] and det(A) = 1.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 598.
M. Petkovsek et al., A=B, Peters, 1996, p. 21.

Antoine Mathys, Table of n, a(n) for n = 0..496 (first 100 terms by T. D. Noe)
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
E. C. Catalan, Théorème de MM. Smith et Mansion, Nouvelle correspondance mathématique, 4 (1878) 103-112. [_Philippe Deléham_, Dec 22 2003]
Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, 76 (2003), 392-394.
P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 81-82.
Mathoverflow, Asymptotics of product of Euler's totient function, 2016.
H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.
Eric Weisstein's World of Mathematics, Le Paige's Theorem
Index to divisibility sequences

Cf. A000010, A060238, A060239, A059381, A059382, A059383, A059384, A002088.

Cf. A003989.

List([1..30],n->Product([1..n],i->Phi(i))); # Muniru A Asiru, Jul 31 2018

a001088 n = a001088_list !! (n-1)
a001088_list = scanl1 (*) a000010_list
-- Reinhard Zumkeller, Mar 04 2012

with(numtheory,phi); A001088 := proc(n) local i; mul(phi(i),i=1..n); end;
seq(A001088(n), n=0..30);

A001088[n_]:=Times@@EulerPhi/@Range[n]; Table[A001088[n], {n, 30}] (* Enrique Pérez Herrero, Sep 19 2010 *)
Rest[FoldList[Times,1,EulerPhi[Range[30]]]] (* Harvey P. Dale, Dec 09 2011 *)

a(n)=prod(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Mar 04 2012

a(n) = phi(1) * phi(2) * ... * phi(n).

Limit_{n->infinity} a(n)^(1/n) / n = exp(-1) * A124175 = 0.205963050288186353879675428232497466485878059342058515016427881513657493... (see Mathoverflow link). - Vaclav Kotesovec, Jun 09 2021

a(0)=1 prepended by Alois P. Heinz, Jul 19 2023

A175836 a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 72, 432, 5184, 41472, 497664, 5971968, 107495424, 1289945088, 30958682112, 433421549568, 10402117189632, 249650812551168, 5991619501228032, 107849151022104576, 3882569436795764736

Offset: 1

Enrique Pérez Herrero, Sep 18 2010

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A060648(gcd(i,j)) for 1 <= i,j <= n, note that A060648 is the Inverse Möbius transform of A001615. - Enrique Pérez Herrero, Aug 12 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..423
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
Index to divisibility sequences

Cf. A001615, A001088, A059381, A059382, A059383, A059384, A238498.

Cf. A239672.

a175836 n = a175836_list !! (n-1)
a175836_list = scanl1 (*) a001615_list
-- 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: # Peter Luschny, Feb 28 2014

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n];
DedekindPsi[n_]:=JordanTotient[n,2]/EulerPhi[n];
A175836[n_]:=Times@@DedekindPsi/@Range[n]

a=direuler(p=2, 100, (1+X)/(1-p*X));for(i=2,#a,a[i]*=a[i-1]);a
\\ Charles R Greathouse IV, Jul 29 2011

a(n) = A059381(n)/A001088(n).

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 *)

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]]

A177066 Determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(2i-1,2j-1) for 1 <= i,j <= n.

Original entry on oeis.org

1, 2, 8, 48, 288, 2880, 34560, 276480, 4423680, 79626240, 955514880, 21021327360, 420426547200, 7567677849600, 211894979788800, 6356849393664000, 127136987873280000, 3051287708958720000, 109846357522513920000

Offset: 1

John W. Layman, Dec 09 2010

nonn

It appears, but has not been proved, that the ratios a(n+1)/a(n) give phi(2n+1) (A037225).

See A001088, A059381, and A059382 for determinants of matrices M defined by M(i,j) = gcd(i,j), gcd(i^2,j^2), and gcd(i^3,j^3), respectively.

Cf. A001088, A037225, A059381, A059382.

A177066 := proc(n) M := Matrix(n) ; for i from 1 to n do for j from 1 to n do M[i,j] := igcd(2*i-1,2*j-1) ; end do: end do: LinearAlgebra[Determinant](M) ; end proc: # R. J. Mathar, Dec 10 2010

cp's OEIS Frontend

A238688 Triangle read by rows: T(n,k) = A059381(n)/(A059381(k)*A059381(n-k)).

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A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

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A175836 a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function (A001615).

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

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

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

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

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