A010786 -id:A010786 - cp's OEIS frontend

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

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

A067514 Number of distinct primes of the form floor(n/k) for 1 <= k <= n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 1, 2, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 5, 6, 2, 3, 4, 4, 4, 5, 4, 5, 3, 4, 5, 6, 4, 5, 5, 6, 4, 5, 4, 5, 5, 5, 6, 7, 3, 4, 4, 5, 6, 7, 5, 6, 5, 6, 7, 8, 4, 5, 5, 5, 4, 5, 6, 7, 7, 8, 7, 8, 4, 5, 5, 5, 5, 6, 7, 8, 6, 6, 7, 8, 4, 5, 6, 7, 7, 8, 5, 6, 7, 8, 9, 10, 6, 7, 5, 6, 5, 6, 6

Offset: 1

Amarnath Murthy, Feb 12 2002

			a(10)=3 as floor(10/k) for k = 1 to 10 is 10,5,3,2,2,1,1,1,1,1, respectively; the 3 primes are 5,3,2.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Randell Heyman, Cardinality of a floor function, arXiv:1905.00533 [math.NT], 2019.
Randell Heyman, Primes in floor function sets, arXiv:2111.00408 [math.NT], 2021
Randell Heyman, Primes in floor function sets, INTEGERS, vol. 22, 2022, A59.

Cf. A068050.

Cf. A055086 (number of distinct integers with same form). - Michel Marcus, May 04 2019

a[n_] := Length[Union[Select[Table[Floor[n/i], {i, 1, n}], PrimeQ]]]
Table[PrimeNu[Product[Floor[n/k], {k, 1, n}]], {n, 1, 100}] (* G. C. Greubel, May 08 2017 *)

a(n) = #select(x->isprime(x), Set(vector(n, k, n\k))); \\ Michel Marcus, May 04 2019

a(n)=my(s=sqrtint(n+1)); sum(k=1,s,isprime(n\k))+primepi(n\s-1) \\ Charles R Greathouse IV, Nov 05 2021

a(n) = A001221(A010786(n)). - Enrique Pérez Herrero, Feb 26 2012

a(n) = 4*n^(1/2)/log(n) + O(n^(1/2)/(log(n))^2). - Randell Heyman, Oct 06 2022

Edited by Dean Hickerson, Feb 12 2002

A317980 a(n) = Product_{i=1..n} floor(5*i/2).

Original entry on oeis.org

2, 10, 70, 700, 8400, 126000, 2142000, 42840000, 942480000, 23562000000, 636174000000, 19085220000000, 610727040000000, 21375446400000000, 790891516800000000, 31635660672000000000, 1328697748224000000000, 59791398670080000000000, 2810195737493760000000000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

If p > 2 and p is odd, then Product_{i=1..n} floor(p*i/2) ~ (p/2)^n * n! * 2^(1/(2*p)) * sqrt(Pi) / (Gamma(1/2 - 1/(2*p)) * n^(1/(2*p))).

Harvey P. Dale, Table of n, a(n) for n = 1..390

Cf. A010786, A180736, A275062, A319948, A319950.

Table[Product[Floor[i*5/2], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{4 a[n] - 10 a[n - 1] - 5 (n - 1) (5 n - 6) a[n - 2] == 0, a[1] == 2, a[2] == 10}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)
FoldList[Times,Floor[5*Range[20]/2]] (* Harvey P. Dale, Sep 17 2020 *)

a(n) ~ (5/2)^n * n! * 2^(1/10) * sqrt(Pi) / (Gamma(2/5) * n^(1/10)).

Recurrence: 4*a(n) - 10*a(n-1) - 5*(n - 1)*(5*n - 6)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319948 a(n) = Product_{i=1..n} floor(3*i/2).

Original entry on oeis.org

1, 3, 12, 72, 504, 4536, 45360, 544320, 7076160, 106142400, 1698278400, 30569011200, 580811212800, 12197035468800, 268334780313600, 6440034727526400, 161000868188160000, 4347023441080320000, 121716656350248960000, 3651499690507468800000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

Cf. A010786, A180736, A275062, A319949, A319950, A317980.

Table[Product[Floor[i*3/2], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{4 a[n] - 6 a[n - 1] - 3 (n - 1) (3 n - 4) a[n - 2] == 0, a[1] == 1, a[2] == 3}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)

a(n) ~ (3/2)^n * n! * 2^(1/6) * sqrt(Pi) / (Gamma(1/3) * n^(1/6)).

Recurrence: 4*a(n) - 6*a(n-1) - 3*(n - 1)*(3*n - 4)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319949 a(n) = Product_{i=1..n} floor(4*i/3).

Original entry on oeis.org

1, 2, 8, 40, 240, 1920, 17280, 172800, 2073600, 26956800, 377395200, 6038323200, 102651494400, 1847726899200, 36954537984000, 776045297664000, 17072996548608000, 409751917166592000, 10243797929164800000, 266338746158284800000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..429

Cf. A004773, A010786, A180736, A275062, A319948, A319950.

Table[Product[Floor[i*4/3], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{27*(3*n - 7)*a[n] == 54*(2*n - 5)*a[n-1] + 12*(12*n^2 - 42*n + 35)*a[n-2] + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a[n-3], a[1]==1, a[2]==2, a[3]==8}, a, {n, 1, 20}]
FoldList[Times,Floor[4 Range[20]/3]] (* Harvey P. Dale, Mar 21 2024 *)

a(n) = prod(i=1, n, (4*i)\3); \\ Michel Marcus, Oct 03 2018

a(n) ~ (4/3)^n * n! * 2*sqrt(Pi) / (3^(1/4) * Gamma(1/4) * n^(1/4)).

Recurrence: 27*(3*n - 7)*a(n) = 54*(2*n - 5)*a(n-1) + 12*(12*n^2 - 42*n + 35)*a(n-2) + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a(n-3).

A319950 a(n) = Product_{i=1..n} floor(5*i/3).

Original entry on oeis.org

1, 3, 15, 90, 720, 7200, 79200, 1029600, 15444000, 247104000, 4447872000, 88957440000, 1868106240000, 42966443520000, 1074161088000000, 27928188288000000, 781989272064000000, 23459678161920000000, 727250023019520000000, 23999250759644160000000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

If p > 3 and gcd(p,3)=1 then Product_{i=1..n} floor(i*p/3) ~ (p/3)^n * n! * 2*Pi * 3^(1/p - 1/2) / (c(p) * n^(1/p)), where
c(p) = Gamma(2/3 - 2/(3*p)) * Gamma(1/3 - 1/(3*p)) if mod(p, 3) = 1,
c(p) = Gamma(1/3 - 2/(3*p)) * Gamma(2/3 - 1/(3*p)) if mod(p, 3) = 2.
In general, if q > 1, p > q and gcd(p,q)=1, then Product_{i=1..n} floor(i*p/q) ~ c(p,q) * (p/q)^n * n! / n^((q-1)/(2*p)), where c(p,q) is a constant.

Cf. A010786, A047220, A180736, A275062, A319948, A319949, A317980.

Table[Product[Floor[i*5/3], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{27*(15*n - 32)*a[n] == 675*(n-2)*a[n-1] + 15*(75*n^2 - 255*n + 194)*a[n-2] + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a[n-3], a[1]==1, a[2]==3, a[3]==15}, a, {n, 1, 20}]

a(n) = prod(i=1, n, (5*i)\3); \\ Michel Marcus, Oct 03 2018

a(n) ~ (5/3)^n * n! * 2*Pi / (3^(3/10) * Gamma(1/5) * Gamma(3/5) * n^(1/5)).

Recurrence: 27*(15*n - 32)*a(n) = 675*(n-2)*a(n-1) + 15*(75*n^2 - 255*n + 194)*a(n-2) + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a(n-3).

A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

Original entry on oeis.org

1, 1, 2, 9, 96, 1250, 64800, 1764735, 224788480, 22499086176, 6123600000000, 408514437465750, 1308805762115174400, 133962125607455951520, 99335199198879310098432, 113040832521732593994140625, 425230288403106927476736000000, 72623663171934137824096600064000

Offset: 0

Vaclav Kotesovec, Jun 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..160

Cf. A010786, A034841, A051054, A092143, A272093.

[n eq 0 select 1 else (&*[Binomial(n,Floor(n/j)): j in [1..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024

Table[Product[Binomial[n, Floor[n/k]], {k, 1, n}], {n, 0, 20}]
Table[Product[((n + 1)/k - 1)^Floor[n/k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 24 2021 *)

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

log(a(n)) ~ n * log(n)^2 / 2. - Vaclav Kotesovec, Jun 21 2021

a(n) = Product_{k=1..n} ((n+1)/k - 1)^floor(n/k). - Vaclav Kotesovec, Jun 24 2021

A076891 a(n) = [n/1][n/2][n/3] ...[n/n] / n^(tau(n)/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 6, 2, 8, 6, 60, 2, 288, 48, 84, 60, 3840, 48, 15552, 144, 4800, 10080, 221760, 96, 331776, 138240, 224640, 25920, 20321280, 4032, 108864000, 345600, 17694720, 36495360, 51701760, 40320, 11287019520, 627056640, 992839680

Offset: 1

Benoit Cloitre, Nov 26 2002

nonn

Cf. A000005 (tau), A007955, A010786.

a[n_] := Product[Floor[n/k], {k, 1, n}] / n^(DivisorSigma[0, n]/2); Array[a, 40] (* Amiram Eldar, Apr 08 2024 *)

a(n) = prod(k = 1, n, n\k) / n^(numdiv(n)/2); \\ Amiram Eldar, Apr 08 2024

a(n) = A010786(n)/n^(A000005(n)/2) = A010786(n)/A007955(n).

A308820 a(n) = Product_{k=1..n} ceiling(n/k)!.

Original entry on oeis.org

1, 2, 12, 96, 2880, 34560, 5806080, 92897280, 25082265600, 2006581248000, 794606174208000, 19070548180992000, 208250386136432640000, 5831010811820113920000, 4198327784510482022400000, 3224315738504050193203200000, 14799609239733590386802688000000

Offset: 1

Ilya Gutkovskiy, Jun 26 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 1..178

Cf. A010786, A020696, A092143, A131385.

[(&*[Factorial(Ceiling(n/(n-j+1))): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Mar 08 2023

seq(mul(ceil(n/k)!, k=1..n), n=1..30); # Ridouane Oudra, Apr 10 2023

a[n_] := Product[Ceiling[n/k]!, {k, 1, n}]; Table[a[n], {n, 1, 17}]

a(n) = prod(k=1, n, ceil(n/k)!); \\ Michel Marcus, Jun 27 2019

def A308820(n): return product( factorial(ceil(n/(n-k+1))) for k in range(1,n+1))
[A308820(n) for n in range(1,21)] # G. C. Greubel, Mar 08 2023

a(n) = Product_{k=1..n-1} Product_{d|k} (d + 1).
a(n) = Product_{k=1..n-1} (k + 1)^floor((n-1)/k). - Ridouane Oudra, Apr 10 2023
a(n) = A131385(n)*A092143(n-1). - Ridouane Oudra, Sep 20 2024

A309912 a(n) = Product_{p prime, p <= n} floor(n/p).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 8, 12, 30, 30, 48, 48, 112, 210, 240, 240, 324, 324, 480, 840, 1848, 1848, 2304, 2880, 6240, 7020, 10080, 10080, 14400, 14400, 15360, 25344, 53856, 78540, 90720, 90720, 191520, 311220, 374400, 374400, 508032, 508032, 709632, 855360, 1788480, 1788480

Offset: 0

Ilya Gutkovskiy, Aug 22 2019

nonn

Product of exponents of prime factorization of A048803 (squarefree factorials).

			A048803(14) = 1816214400 = 2^7 * 3^4 * 5^2 * 7^2 * 11 * 13 so a(14) = 7 * 4 * 2 * 2 * 1 * 1 = 112.

Cf. A000040, A000720, A005361, A010786, A013939, A048803, A135291.

a:= n-> mul(floor(n/p), p=select(isprime, [$2..n])):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 23 2019

Table[Product[Floor[n/Prime[k]], {k, 1, PrimePi[n]}], {n, 0, 47}]

from math import prod
from sympy import primerange
def A309912(n): return prod(n//p for p in primerange(n)) # Chai Wah Wu, Jun 02 2025

a(n) = Product_{k=1..A000720(n)} floor(n/A000040(k)).

a(n) = A005361(A048803(n)).

cp's OEIS Frontend

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

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A067514 Number of distinct primes of the form floor(n/k) for 1 <= k <= n.

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A317980 a(n) = Product_{i=1..n} floor(5*i/2).

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A319948 a(n) = Product_{i=1..n} floor(3*i/2).

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A319949 a(n) = Product_{i=1..n} floor(4*i/3).

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A319950 a(n) = Product_{i=1..n} floor(5*i/3).

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A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

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A076891 a(n) = [n/1][n/2][n/3] ...[n/n] / n^(tau(n)/2).

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