A117871 -id:A117871 - cp's OEIS frontend

A129635 Decimal expansion of 1 + A117871.

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

Offset: 1

Jonathan Vos Post, May 31 2007

			2.6917992098217123513392618067876318698236937629258191345569...

Cf. A092143. See A117871 for a better version.

More terms from R. J. Mathar, Sep 02 2007

Edited by N. J. A. Sloane, Sep 16 2007 and May 06 2008

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

Original entry on oeis.org

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

A345683 a(n) = n! * Sum_{k=1..n} 1/floor(n/k).

Original entry on oeis.org

1, 3, 14, 66, 444, 2880, 25080, 216720, 2247840, 24071040, 304335360, 3752179200, 54965433600, 810550540800, 13176376012800, 219079045785600, 4078723532083200, 75227891042304000, 1550619342784512000, 31871016307113984000, 710529031487987712000, 16180987966182014976000

Offset: 1

Vaclav Kotesovec, Jun 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..448
Vaclav Kotesovec, Plot of a(n) / (n*n!) for n = 1..1000000
Mathoverflow, An asymptotic formula for a sum involving powers of floor functions, 2019.

Cf. A006218, A010786, A092143, A117871, A345682, A345684, A345686, A355987.

Table[n! * Sum[1/Floor[n/k], {k, 1, n}], {n, 1, 25}]
Table[n!*(Sum[(Floor[n/j] - Floor[n/(j + 1)])/j, {j, 1, n}]), {n, 1, 25}]

a(n) = n!*sum(k=1, n, 1/(n\k)); \\ Michel Marcus, Jun 24 2021

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (1-x^k)*log(1-x^k))/(1-x))) \\ Seiichi Manyama, Jul 23 2022

from math import factorial, isqrt
def A345683(n): return (m:=factorial(n))*(n-1)+m//n+sum((q:=n//k)*(m//k-m//(k-1))+m//q for k in range(2,isqrt(n)+1)) # Chai Wah Wu, Oct 27 2023

a(n) ~ c * n * n!, where c = Sum_{j>=1} 1/(j^2*(j+1)) = Pi^2/6 - 1 = 0.644934... [proved by Harry Richman, see Mathoverflow link]

E.g.f.: -(1/(1-x)) * Sum_{k>0} (1 - x^k) * log(1 - x^k). - Seiichi Manyama, Jul 23 2022

A345682 a(n) = n! * Sum_{k=1..n} 1/(k*floor(n/k)).

Original entry on oeis.org

1, 2, 7, 26, 148, 804, 6228, 47424, 441936, 4288320, 50437440, 560373120, 7723935360, 106618256640, 1614841401600, 25127582054400, 446784010444800, 7727747269939200, 152873884406476800, 2966599550251008000, 62987912790921216000, 1378192085174919168000

Offset: 1

Vaclav Kotesovec, Jun 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..449
Vaclav Kotesovec, Plot of a(n)/n! for n = 1..1000000

Cf. A006218, A010786, A024917, A092143, A117871, A345683, A345684.

Table[n! * Sum[1/(k*Floor[n/k]), {k, 1, n}], {n, 1, 25}]
Table[n! * Sum[(HarmonicNumber[Floor[n/j]] - HarmonicNumber[Floor[n/(1 + j)]])/j, {j, 1, n}], {n, 1, 25}]

a(n) = n!*sum(k=1, n, 1/(k*(n\k))); \\ Michel Marcus, Jun 24 2021

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (1-x^k)*log(1-x^k)/k)/(1-x))) \\ Seiichi Manyama, Jul 23 2022

a(n) ~ c * n!, where c = Sum_{j>=1} log(1 + 1/j)/j = A131688 = 1.25774...

E.g.f.: -(1/(1-x)) * Sum_{k>0} (1 - x^k) * log(1 - x^k)/k. - Seiichi Manyama, Jul 23 2022

A345684 a(n) = n! * Sum_{k=1..n} k/floor(n/k).

Original entry on oeis.org

1, 5, 32, 198, 1584, 12480, 122520, 1214640, 14011200, 166924800, 2274894720, 31135104000, 485667705600, 7710089587200, 133974352512000, 2386854434764800, 46621903994265600, 918384939343872000, 19760215067873280000, 430137075045629952000, 10042411264251125760000

Offset: 1

Vaclav Kotesovec, Jun 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..448

Cf. A006218, A010786, A076000, A092143, A117871, A345682, A345683.

Table[n!*Sum[k/Floor[n/k], {k, 1, n}], {n, 1, 25}]
Table[n!*Sum[(Floor[n/j] - Floor[n/(1 + j)])*((1 + Floor[n/j] + Floor[n/(1 + j)])/2/j), {j, 1, n}], {n, 1, 25}]

a(n) = n!*sum(k=1, n, k/(n\k)); \\ Michel Marcus, Jun 23 2021

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k*(1-x^k)*log(1-x^k))/(1-x))) \\ Seiichi Manyama, Jul 23 2022

a(n) ~ c * n^2 * n!, where c = Sum_{j>=1} (2*j + 1) / (2*j^3*(j+1)^2) = Pi^2/12 + zeta(3)/2 - 1 = 0.423495...

E.g.f.: -(1/(1-x)) * Sum_{k>0} k * (1 - x^k) * log(1 - x^k). - Seiichi Manyama, Jul 23 2022

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A129635 Decimal expansion of 1 + A117871.

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

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A345683 a(n) = n! * Sum_{k=1..n} 1/floor(n/k).

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A345682 a(n) = n! * Sum_{k=1..n} 1/(k*floor(n/k)).

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A345684 a(n) = n! * Sum_{k=1..n} k/floor(n/k).

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