A092287 -id:A092287 - cp's OEIS frontend

A129365 a(n) = A092287(n)/A129364(n).

1, 1, 1, 1, 1, 2, 2, 2, 6, 48, 48, 48, 48, 1536, 207360, 207360, 207360, 1105920, 1105920, 17694720, 30098718720, 15410543984640, 15410543984640, 481579499520, 60197437440000, 123284351877120000, 29958097506140160000

Offset: 1

Peter Bala, Apr 13 2007

Conjectures:
A) a(n) is always an integer.
B) If p is a prime then p|a(n) if and only if p <= n/3. Let ordp(n,p) denote the exponent of the largest power of p which divides n. For example, ordp(48,2) = 4 since 48 = 3*(2^4). The precise decomposition of a(n) into primes would follow from the following two conjectures:
C) For each positive integer n and prime p, ordp(a(n*p),p) = ordp(a(n*p+1),p) = ordp(a(n*p+2),p) = . . . = ordp(a(n*p+p-1),p).
D) Let b(n) = A004125(n). Then ordp(a(n*p),p) = b(n) + b(floor(n/p)) + b(floor(n/p^2)) + b(floor(n/p^3)) + .... This is reminiscent of de Polignac's formula (also due to Legendre) for the prime factorization of n! (see the link).

Wikipedia, De Polignac's formula.

Cf. A004125, A092287, A129364.

a(n) = ( Product_{j = 1..n} Product_{k = 1..n} gcd(j,k) ) / ( Product_{j = 1..n} Product_{d|j} d^(j/d) ).

a(n) = ( Product_{j = 1..n} Product_{k = 1..n} gcd(j,k) ) / ( Product_{k = 1..n} (floor(n/k)!)^k ).

A129453 An analog of Pascal's triangle based on A092287. T(n,k) = A092287(n)/(A092287(n-k)*A092287(k)), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 16, 24, 16, 1, 1, 5, 40, 40, 5, 1, 1, 864, 2160, 11520, 2160, 864, 1, 1, 7, 3024, 5040, 5040, 3024, 7, 1, 1, 2048, 7168, 2064384, 645120, 2064384, 7168, 2048, 1, 1, 729, 746496, 1741824, 94058496, 94058496, 1741824, 746496, 729, 1

Offset: 0

Peter Bala, Apr 16 2007

It appears that the T(n,k) are always integers. This would follow from the conjectured prime factorization given in A092287. Calculation suggests that the binomial coefficients C(n,k) divide T(n,k) and indeed that T(n,k)/C(n,k) are perfect squares.

			Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1, 16, 24, 16,  1;
  1,  5, 40, 40,  5,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A092287, A129454, A129455.

A092287:= func< n | n eq 0 select 1 else (&*[(&*[GCD(j,k): k in [1..n]]): j in [1..n]]) >;
A129453:= func< n,k | A092287(n)/(A092287(n-k)*A092287(k)) >;
[A129453(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2024

A092287[n_]:= Product[GCD[j,k], {j,n}, {k,n}];
A129453[n_, k_]:= A092287[n]/(A092287[k]*A092287[n-k]);
Table[A129453[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2024 *)

def A092287(n): return product(product( gcd(j,k) for k in range(1,n+1)) for j in range(1,n+1))
def A129453(n,k): return A092287(n)/(A092287(n-k)*A092287(k))
flatten([[A129453(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 07 2024

T(n, k) = (Product_{i=1..n} Product_{j=1..n} gcd(i,j)) / ( (Product_{i=1..n-k} Product_{j=1..n-k} gcd(i,j)) * ( Product_{i=1..k} Product_{j=1..k} gcd(i,j)) ), note that empty products equal to 1.

T(n, n-k) = T(n, k). - G. C. Greubel, Feb 07 2024

A051190 a(n) = Product_{k=1..n-1} gcd(k,n).

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 16, 9, 80, 1, 3456, 1, 448, 2025, 2048, 1, 186624, 1, 1024000, 35721, 11264, 1, 573308928, 625, 53248, 59049, 179830784, 1, 1007769600000, 1, 67108864, 7144929, 1114112, 37515625, 160489808068608, 1, 4980736, 89813529

Offset: 1

Antti Karttunen, Oct 21 1999

a(n) > 1 if and only if n is composite. - Charles R Greathouse IV, Jan 04 2013

T. D. Noe, Table of n, a(n) for n = 1..200
OEIS Wiki, Generalizations of the factorial

Cf. A067911, A006579, A224479, A092287.

a051190 n = product $ map (gcd n) [1..n-1]
-- Reinhard Zumkeller, Nov 22 2011

A051190 := proc(n) local i; mul(igcd(n, i ), i = 1..(n-1)) end;

a[n_] := If[PrimeQ[n], 1, Times @@ (GCD[n, #]& /@ Range[n-1])]; Table[a[n], {n, 1, 39}]  (* Jean-François Alcover, Jul 18 2012 *)
Table[Times @@ GCD[n, Range[n-1]], {n, 50}] (* T. D. Noe, Apr 12 2013 *)
Table[Product[GCD[k,n],{k,n-1}],{n,50}] (* Harvey P. Dale, Jan 29 2025 *)

a(n)=my(f=factor(n)); prod(i=1, #f[,1], prod(j=1, f[i,2], f[i,1]^(n\f[i,1]^j)))/n \\ Charles R Greathouse IV, Jan 04 2013

a(n) = prod(k=1,n-1,gcd(k,n)); /* Joerg Arndt, Apr 14 2013 */

A051190 = lambda n: mul(gcd(n,i) for i in (1..n-1))
[A051190(n) for n in (1..39)] # Peter Luschny, Apr 07 2013

# A second, faster version, based on the prime factorization of a(n):
def A051190(n):
    R = 1
    if not is_prime(n) :
        for p in primes(n//2+1):
            s = 0; r = n; t = n-1
            while r > 0 :
                r = r//p; t = t//p
                s += (r-t)*(r+t-1)
            R *= p^(s/2)
    return R
[A051190(i) for i in (1..1000)]  # Peter Luschny, Apr 08 2013

a(n) = Product_{ d divides n, d < n } d^phi(n/d). - Peter Luschny, Apr 07 2013
a(n) = A067911(n) / n. - Peter Luschny, Apr 07 2013
Product_{j=1..n} Product_{k=1..j-1} gcd(j,k), n >= 1. - Daniel Forgues, Apr 11 2013
a(n) = sqrt( (1/n) * (A092287(n) / A092287(n-1)) ). - Daniel Forgues, Apr 13 2013

A129454 a(n) = Product{i=1..n-1} Product{j=1..n-1} Product{k=1..n-1} gcd(i,j,k).

Original entry on oeis.org

1, 1, 1, 2, 6, 1536, 7680, 8806025134080, 61642175938560, 2168841254587541957294161920, 7562281854741110985626291951024209920, 1362299589723309231779453337910253309054734620740812800000000

Offset: 0

Peter Bala, Apr 16 2007

nonn

Conjecture: Let p be a prime and let ordp(n,p) denote the largest power of p which divides n. For example, ordp(48,2)=4 since 48 = 3*(2^4). Then we conjecture that the prime factorization of a(n) is given by ordp(a(n),p)=(floor(n/p))^3 + (floor(n/p^2))^3 + (floor(n/p^3))^3 + . . .. Compare with the comments in A092287.

G. C. Greubel, Table of n, a(n) for n = 0..26

Cf. A092287, A129455.

A129454:= func< n | n le 1 select 1 else (&*[(&*[(&*[GCD(GCD(j,k),m): k in [1..n-1]]): j in [1..n-1]]): m in [1..n-1]]) >;
[A129454(n): n in [0..20]]; // G. C. Greubel, Feb 07 2024

A129454[n_]:= Product[GCD[j,k,m], {j,n-1}, {k,n-1}, {m,n-1}];
Table[A129454[n], {n,0,20}] (* G. C. Greubel, Feb 07 2024 *)

def A129454(n): return product(product(product(gcd(gcd(j,k),m) for k in range(1,n)) for j in range(1,n)) for m in range(1,n))
[A129454(n) for n in range(21)] # G. C. Greubel, Feb 07 2024

a(n) = Product{i=1..n-1} Product{j=1..n-1} Product{k=1..n-1} gcd(i,j,k), for n > 2, otherwise a(n) = 1.

A129455 An analog of Pascal's triangle based on A129454. T(n, k) = A129454(n+1)/(A129454(n-k+1)*A129454(k+1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 256, 384, 256, 1, 1, 5, 640, 640, 5, 1, 1, 1146617856, 2866544640, 244611809280, 2866544640, 1146617856, 1, 1, 7, 4013162496, 6688604160, 6688604160, 4013162496, 7, 1, 1, 35184372088832, 123145302310912, 47066867504069920948224, 919274755938865643520, 47066867504069920948224, 123145302310912, 35184372088832, 1

Offset: 0

Peter Bala, Apr 16 2007

It appears that the T(n,k) are always integers. This would follow from the conjectured prime factorization given in A129454. Calculation suggests that the binomial coefficients C(n,k) divide T(n,k) and that T(n,k)/C(n,k) are perfect sixth powers.

			Triangle starts:
  1;
  1,   1;
  1,   2,   1;
  1,   3,   3,   1;
  1, 256, 384, 256,  1;
  1,   5, 640, 640,  5,  1;

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Cf. A092287, A129453, A129454.

A129454:= func< n | n le 1 select 1 else (&*[(&*[(&*[GCD(GCD(j,k),m): k in [1..n-1]]): j in [1..n-1]]): m in [1..n-1]]) >;
A129455:= func< n,k | A129454(n+1)/(A129454(n-k+1)*A129454(k+1)) >;
[A129455(n,k): k in [0..n], n in [0..9]]; // G. C. Greubel, Feb 07 2024

A129454[n_]:= Product[GCD[j,k,m], {j,n-1}, {k,n-1}, {m,n-1}];
A129455[n_, k_]:= A129454[n+1]/(A129454[k+1]*A129454[n-k+1]);
Table[A129455[n,k], {n,0,9}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2024 *)

def A129454(n): return product(product(product(gcd(gcd(j,k),m) for k in range(1,n)) for j in range(1,n)) for m in range(1,n))
def A129455(n,k): return A129454(n+1)/(A129454(n-k+1)*A129454(k+1))
flatten([[A129455(n,k) for k in range(n+1)] for n in range(10)]) # G. C. Greubel, Feb 07 2024

T(n, k) = Product_{h=1..n} Product_{i=1..n} Product_{j=1..n} gcd(h,i,j)/( (Product_{h=1..n-k} Product_{i=1..n-k} Product_{j=1..n-k} gcd(h,i,j))*(Product_{h=1..k} Product_{i=1..k} Product_{j=1..k} gcd(h,i,j)) ).

T(n, n-k) = T(n, k). - G. C. Greubel, Feb 07 2024

A224479 a(n) = Product_{k=1..n} Product_{i=1..k-1} gcd(k,i).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 24, 24, 384, 3456, 276480, 276480, 955514880, 955514880, 428070666240, 866843099136000, 1775294667030528000, 1775294667030528000, 331312591939905257472000, 331312591939905257472000, 339264094146462983651328000000

Offset: 0

Peter Luschny, Apr 07 2013

nonn

The order of the primes in the prime factorization of a(n) is given by
ord_{p}(a(n)) = (1/2)*Sum_{i>=1} floor(n/p^i)*(floor(n/p^i)-1).
Product of all entries of lower-left (excluding main diagonal) triangular submatrix of GCDs. Also the product of all entries of upper-right (excluding main diagonal) triangular submatrix of GCDs, since the matrix is symmetric. - Daniel Forgues, Apr 14 2013
a(n)^2 * n! gives A092287(n), where n! is the product of the main diagonal entries of the matrix. - Daniel Forgues, Apr 14 2013

Charles R Greathouse IV, Table of n, a(n) for n = 0..97
OEIS Wiki, Generalizations of the factorial

Cf. A051190, A092287.

A224479 := proc(n) local h, k, d;
mul(mul(d^phi(k/d), d = divisors(k) minus {k}), k = 1..n) end:
seq(A224479(i), i = 0..20);

a[n_] := Product[ d^EulerPhi[k/d], {k, 1, n}, {d, Divisors[k] // Most}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 27 2013, after Maple *)

a(n)=prod(k=1,n,my(s=1);fordiv(k,d,dCharles R Greathouse IV, Jun 27 2013

def A224479(n):
    R = 1;
    for p in primes(n):
        s = 0; r = n
        while r > 0 :
            r = r//p
            s += r*(r-1)
        R *= p^(s/2)
    return R
[A224479(i) for i in (0..20)]

a(n) = Product_{k=1..n} Product_{d divides k, d < k} d^phi(k/d).
n! * a(n)^2 = A092287(n).
a(n)/a(n-1) = A051190(n) for n >= 1.
a(n) = sqrt(A092287(n) / n!). - Daniel Forgues, Apr 14 2013

A129364 a(n) = Product_{k = 1..n} A066841(k).

Original entry on oeis.org

1, 2, 6, 96, 480, 207360, 1451520, 2972712960, 722369249280, 5778953994240000, 63568493936640000, 9111096278347394580480000, 118444251618516129546240000, 10400352846118664303196241920000

Offset: 1

Peter Bala, Apr 11 2007

Conjecture: a(n) divides A092287(n) for all n - see comments in A129365.

Cf. A007955, A066841, A092143, A092287, A129365.

Table[Product[Floor[n/k]!^k, {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Jun 24 2021 *)
Table[Product[k^(Floor[n/k]*(1 + Floor[n/k])/2), {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Jun 24 2021 *)

a(n) = prod(k=1, n, k^((n\k) * (1 + n\k) \ 2)); \\ Daniel Suteu, Sep 12 2018

a(n) = Product_{k = 1..n} Product_{d|k} d^(k/d).
a(n) = Product_{k = 1..n} ((floor(n/k))!)^k.
a(n) = exp(Sum_{k = 1..n} log(k)/2 * floor(n/k) * floor(1 + n/k)). - Daniel Suteu, Sep 12 2018
log(a(n)) ~ c * n^2, where c = -zeta'(2)/2 = A073002/2 = 0.468774... - Vaclav Kotesovec, Jun 24 2021

A224497 a(n) = sqrt(floor(n/2)! * Product_{k=1..n} Product_{i=1..k-1} gcd(k,i)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 12, 12, 96, 288, 5760, 5760, 829440, 829440, 46448640, 2090188800, 267544166400, 267544166400, 346737239654400, 346737239654400, 1109559166894080000, 209706682542981120000, 73816752255129354240000, 73816752255129354240000

Offset: 0

Peter Luschny, Apr 08 2013

nonn

The order of the primes in the prime factorization of a(n) is given by
ord_{p}(a(n)) = (1/4)*Sum_{i>=1} floor(n/p^i)*(floor(n/p^i)-1) + (1/2)*Sum_{i>=1} floor(floor(n/2)/p^i).
For n > 1: a(n) = a(n-1) if and only if n is prime.

Cf. A224479.

A224497 := n -> sqrt(iquo(n,2)!*mul(mul(igcd(k,i), i=1..k-1), k=1..n)):
seq(A224497(i), i = 0..23);

def A224497(n):
    R = 1;
    for p in primes(n):
        s = 0; t = 0
        r = n; u = n//2
        while r > 0 :
            r = r//p; u = u//p
            t += u; s += r*(r-1)
        R *= p^((t+s/2)/2)
    return R
[A224497(i) for i in (0..23)]

a(n) = sqrt(floor(n/2)! * A224479(n)).

A092287(n) = A056040(n) * a(n)^4.

A090494 Product_{j=1..n} Product_{k=1..n} lcm(j,k).

Original entry on oeis.org

1, 1, 8, 7776, 1146617856, 1289945088000000000, 46798828032806092800000000000, 2350577043461005964030008507760640000000000000, 8206262459636402163263383676462776103575725539328000000000000000, 2746781358330240881921653545637784861521126603512175621574459373964492800000000000000000

Offset: 0

N. J. A. Sloane, Feb 03 2004

nonn

Cf. A018806.

Cf. A092143, A092287, A129365, A129454.

f := n->mul(mul(lcm(j,k),k=1..n),j=1..n);

Let p be a prime and let ordp(n,p) denote the exponent of the largest power of p which divides n. For example, ordp(48,2)=4 since 48 = 3*(2^4). Then the prime factorization of a(n) appears to be given by the formula ordp(a(n),p)= sum_{k >= 1} [(2*(p^k)-1)*floor((n/(p^k)))^2] + 2*sum_{k >= 1} [floor(n/(p^k))*mod(n,p^k)], for each prime p. See the comments sections of A092143, A092287, A129365 and A129454 for similar conjectural prime factorizations. - Peter Bala, Apr 23 2007

A276162 Square array read by antidiagonals: T(n,k) = Product_{i = 1..k} gcd(n, i).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 4, 1, 1, 1, 2, 3, 4, 1, 1, 2, 1, 8, 3, 8, 1, 1, 1, 6, 1, 8, 9, 8, 1, 1, 2, 1, 12, 5, 16, 9, 16, 1, 1, 1, 2, 1, 12, 5, 16, 9, 16, 1, 1, 2, 3, 8, 1, 72, 5, 64, 27, 32, 1, 1, 1, 2, 3, 8, 1, 72, 5, 64, 27, 32, 1, 1, 2, 1, 4

Offset: 1

Peter Kagey, Aug 22 2016

			T(6, 3) = gcd(6, 1) * gcd(6, 2) * gcd(6, 3) = 6.

Peter Kagey, Table of n, a(n) for n = 1..10000
Mohamed Abobakr, Greatest common divisor sequence

Cf. A051190, A067911, A092287.

a276162T n k = product $ map (gcd n) [1..k]
-- Peter Kagey, Aug 23 2016

T(n,k)=prod(i=2,k,gcd(n,i))
for(s=1,15,for(k=1,s-1, print1(T(s-k,k)", "))) \\ Charles R Greathouse IV, Aug 22 2016

cp's OEIS Frontend

A129365 a(n) = A092287(n)/A129364(n).

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A129453 An analog of Pascal's triangle based on A092287. T(n,k) = A092287(n)/(A092287(n-k)*A092287(k)), 0 <= k <= n.

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A051190 a(n) = Product_{k=1..n-1} gcd(k,n).

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A129454 a(n) = Product{i=1..n-1} Product{j=1..n-1} Product{k=1..n-1} gcd(i,j,k).

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A129455 An analog of Pascal's triangle based on A129454. T(n, k) = A129454(n+1)/(A129454(n-k+1)*A129454(k+1)).

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A224479 a(n) = Product_{k=1..n} Product_{i=1..k-1} gcd(k,i).

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A129364 a(n) = Product_{k = 1..n} A066841(k).

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