A280714 -id:A280714 - cp's OEIS frontend

A092143 Cumulative product of all divisors of 1..n.

1, 2, 6, 48, 240, 8640, 60480, 3870720, 104509440, 10450944000, 114960384000, 198651543552000, 2582470066176000, 506164132970496000, 113886929918361600000, 116620216236402278400000, 1982543676018838732800000, 11562194718541867489689600000, 219681699652295482304102400000

Offset: 1

Jon Perry, Mar 31 2004

nonn

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). Let b(n) = A006218(n) = Sum_{k=1..n} floor(n/k). The prime factorization of a(n) appears to be given by the following conjectural formula: ordp(a(n),p) = b(floor(n/p)) + b(floor(n/p^2)) + b(floor(n/p^3)) + ... . Compare with the comments in A129365. - Peter Bala, Apr 15 2007

			a(6) = 1*2*3*2*4*5*2*3*6 = 8640.

G. C. Greubel, Table of n, a(n) for n = 1..190
Angelo B. Mingarelli, Abstract factorials, arXiv:0705.4299 [math.NT], 2007-2012.

Cf. A006218, A010786, A092144, A117871.
Cf. A129364, A129365, A129439, A345466.
Cf. A000005, A000178, A007955, A175493, A280714.

[(&*[j^Floor(n/j): j in [1..n]]): n in [1..30]]; // G. C. Greubel, Feb 05 2024

seq(sqrt(mul(k^numtheory[tau](k), k=1..n)), n=1..40); # Ridouane Oudra, Oct 31 2024

Reap[For[n = k = 1, k <= 25, k++, Do[n = n*d, {d, Divisors[k]}]; Sow[n]]][[2, 1]] (* Jean-François Alcover, Oct 30 2012 *)
Table[Product[k^Floor[n/k], {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Jun 24 2021 *)
FoldList[Times, Times @@@ Divisors[Range[25]]] (* Paolo Xausa, Nov 06 2024 *)

my(z=1); for(i=1,25, fordiv(i,j,z*=j); print1(z, ", "))

[product(j^(n//j) for j in range(1,n+1)) for n in range(1,31)] # G. C. Greubel, Feb 05 2024

a(n) = Product_{k=1..n} {floor(n/k)}!. This formula is due to Sebastian Martin Ruiz. - Peter Bala, Apr 15 2007; Formula corrected by R. J. Mathar, May 06 2008
Sum_{n>=1} 1/a(n) = A117871. - Amiram Eldar, Nov 17 2020
log(a(n)) ~ n * log(n)^2 / 2. - Vaclav Kotesovec, Jun 20 2021
a(n) = Product_{k=1..n} k^floor(n/k). - Vaclav Kotesovec, Jun 24 2021
From Ridouane Oudra, Oct 31 2024: (Start)
a(n) = Product_{k=1..n} A007955(k).
a(n) = Product_{k=1..n} k^(tau(k)/2).
a(n) = sqrt(A175493(n)). (End)
a(n) = A000178(n)/A280714(n). - Amiram Eldar, Aug 16 2025

A280713 Partial sums of A055067 where A055067(n) is the product of non-divisors of n.

Original entry on oeis.org

1, 2, 4, 7, 31, 51, 771, 1401, 14841, 51129, 3679929, 3957129, 482958729, 927745929, 6739632009, 27172044009, 20949961932009, 22047762636009, 6424421468364009, 6728534219386089, 122581010799226089, 2444896564058746089, 1126445624341666426089

Offset: 1

Jaroslav Krizek, Jan 07 2017

nonn

A055067(n) = the product of non-divisors of n.

Paolo Xausa, Table of n, a(n) for n = 1..400

Cf. A055067, A280714.

[&+[Factorial(k) / &*[d: d in Divisors(k)]: k in [1..n]]: n in [1..100]];

Accumulate[Array[#!/Times@@Divisors[#] &, 30]]

a(n) = Sum_{i=1..n} A055067(i).

cp's OEIS Frontend

A092143 Cumulative product of all divisors of 1..n.

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