A048803 -id:A048803 - cp's OEIS frontend

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

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)).

A368092 a(n) = A160014(m, n) * a(n - 1) for m = 2 and n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 9, 135, 405, 8505, 127575, 382725, 1148175, 189448875, 3978426375, 155158628625, 2327379429375, 6982138288125, 20946414864375, 37389350532909375, 112168051598728125, 6393578941127503125, 1054940525286038015625, 3164821575858114046875, 66461253093020394984375

Offset: 0

Peter Luschny, Dec 12 2023

nonn

A160014 are the generalized Clausen numbers. For m = 0 the formula computes the cumulative radical A048803, for m = 1 the Hirzebruch numbers A091137.

Cf. A160014, A048803 (m=0), A091137 (m=1), this sequence (m=2), A368093 (array), A368048, A368117.

from functools import cache
@cache
def a_rec(n):
    if n == 0: return 1
    p = mul(s for s in map(lambda i: i + 2, divisors(n)) if is_prime(s))
    return p * a_rec(n - 1)
print([a_rec(n) for n in range(21)])
# Alternatively, but less efficient:
def a(n): return (2**(n%2 - n) * lcm(product(r + 2 for r in p) for p in Partitions(n)))

a(n) = 2^(n mod 2 - n)*lcm_{p in Partitions(n)} (Product_{t in p}(t + 2)).
a(n) = 2^(n mod 2 - n)*A368048(n).
a(n) = A368117(n) * a(n-1) for n > 0.

A368093 Cumulative products of the generalized Clausen numbers. Array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 12, 6, 1, 1, 9, 24, 12, 1, 5, 5, 135, 720, 60, 1, 1, 25, 5, 405, 1440, 360, 1, 7, 7, 875, 175, 8505, 60480, 2520, 1, 1, 49, 7, 4375, 175, 127575, 120960, 5040, 1, 1, 1, 343, 49, 21875, 875, 382725, 3628800, 15120

Offset: 0

Peter Luschny, Dec 12 2023

A160014 are the generalized Clausen numbers, for m = 0 the formula computes the cumulative radical A048803, and for m = 1 the Hirzebruch numbers A091137.

			Array A(m, n) starts:
  [0] 1, 1,  2,   6,   12,     60,     360,      2520, ...  A048803
  [1] 1, 2, 12,  24,  720,   1440,   60480,    120960, ...  A091137
  [2] 1, 3,  9, 135,  405,   8505,  127575,    382725, ...  A368092
  [3] 1, 1,  5,   5,  175,    175,     875,       875, ...
  [4] 1, 5, 25, 875, 4375,  21875,  765625,  42109375, ...
  [5] 1, 1,  7,   7,   49,     49,    3773,      3773, ...
  [6] 1, 7, 49, 343, 2401, 184877, 1294139, 117766649, ...
  [7] 1, 1,  1,   1,   11,     11,     143,       143, ...
  [8] 1, 1,  1,  11,   11,    143,    1573,      1573, ...
  [9] 1, 1, 11,  11, 1573,   1573,   17303,     17303, ...

Cf. A160014, A048803 (m=0), A091137 (m=1), A368092 (m=2).

Cf. A171080, A238963, A368116, A368048.

from functools import cache
def Clausen(n, k):
    return mul(s for s in map(lambda i: i+n, divisors(k)) if is_prime(s))
@cache
def CumProdClausen(m, n):
    return Clausen(m, n) * CumProdClausen(m, n - 1) if n > 0 else 1
for m in range(10): print([m], [CumProdClausen(m, n) for n in range(8)])

A(m, n) = A160014(m, n) * A(m, n - 1) for n > 0 and A(m, 0) = 1.

A387126 Triangle read by rows: T(n, k) = (n! / (n - k)!) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

1, 1, 1, 2, 4, 4, 6, 18, 36, 36, 12, 48, 144, 288, 288, 60, 300, 1200, 3600, 7200, 7200, 360, 2160, 10800, 43200, 129600, 259200, 259200, 2520, 17640, 105840, 529200, 2116800, 6350400, 12700800, 12700800, 5040, 40320, 282240, 1693440, 8467200, 33868800, 101606400, 203212800, 203212800

Offset: 0

Peter Luschny, Aug 18 2025

			Triangle begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     4,      4;
  [3]    6,    18,     36,     36;
  [4]   12,    48,    144,    288,     288;
  [5]   60,   300,   1200,   3600,    7200,    7200;
  [6]  360,  2160,  10800,  43200,  129600,  259200,   259200;
  [7] 2520, 17640, 105840, 529200, 2116800, 6350400, 12700800, 12700800;

Cf. A007947 (radical), A008279, A048803 (column 0), A277174 (main diagonal).

A387126 := (n, k) -> mul(NumberTheory:-Radical(j), j = 1..n) * n! / (n - k)!:

A387126[n_, k_] := Pochhammer[n-k+1, k] Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
Table[A387126[n, k], {n, 0, 8}, {k, 0, n}]  // Flatten

T(n, k) = A048803(n) * A008279(n, k).

A387138 Triangle read by rows: T(n, k) = binomial(n, k) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 12, 48, 72, 48, 12, 60, 300, 600, 600, 300, 60, 360, 2160, 5400, 7200, 5400, 2160, 360, 2520, 17640, 52920, 88200, 88200, 52920, 17640, 2520, 5040, 40320, 141120, 282240, 352800, 282240, 141120, 40320, 5040

Offset: 0

Peter Luschny, Aug 18 2025

			Triangle begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     4,      2;
  [3]    6,    18,     18,      6;
  [4]   12,    48,     72,     48,     12;
  [5]   60,   300,    600,    600,    300,     60;
  [6]  360,  2160,   5400,   7200,   5400,   2160,    360;
  [7] 2520, 17640,  52920,  88200,  88200,  52920,  17640,  2520;
  [8] 5040, 40320, 141120, 282240, 352800, 282240, 141120, 40320, 5040;

Cf. A007318 (binomial), A007947 (radical), A048803 (column 0 and main diagonal), A387139 (row sums), A387126.

A387138 := (n, k) -> binomial(n, k) * mul(NumberTheory:-Radical(j), j = 1..n):

A387138[n_, k_] := Binomial[n, k] Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
Table[A387138[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = A048803(n) * A007318(n, k).

A387141 a(n) = floor((Product_{k=1..n} radical(k))^(1/n)) for n >= 1, a(0) = 1, where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 13, 14, 14, 14, 15, 15, 15, 16, 15, 16, 16

Offset: 0

Peter Luschny, Aug 18 2025

nonn

Cf. A007947, A048803, A387140.

a := n -> if n = 0 then 1 else floor(mul(NumberTheory:-Radical(k), k = 1..n)^(1/n)) fi:

A387141[n_] := If[n == 0, 1, Floor[Power[Times @@ ResourceFunction["IntegerRadical"][Range[1, n]], 1/n]]]; Table[A387141[n], {n, 0, 74}]

a(n) = floor(A048803(n)^(1/n)) for n >= 1.

A068625 Reduced root factorial of n: product of the smallest integer root of numbers from 1 to n.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 360, 2520, 5040, 15120, 151200, 1663200, 19958400, 259459200, 3632428800, 54486432000, 108972864000, 1852538688000, 33345696384000, 633568231296000, 12671364625920000, 266098657144320000, 5854170457175040000, 134645920515025920000

Offset: 0

Amarnath Murthy, Feb 26 2002

A "binomial" style a(m+n)/(a(m)*a(n)) is not always an integer, as for instance at m = n = 18 (unlike ordinary factorials or A048803). - Hal M. Switkay, Jul 22 2024

			a(8) = 1*2*3*2*5*6*7*2 = 5040.

Alois P. Heinz, Table of n, a(n) for n = 0..462 (first 37 terms from Hal M. Switkay)

Partial products of A052410.

Cf. A000142, A048803.

b:= proc(n) option remember; (l-> (t-> mul(i[1]^(i[2]/t),
       i=l))(igcd(seq(i[2], i=l))))(ifactors(n)[2])
    end:
a:= proc(n) option remember; `if`(n<1, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 22 2024

a(0)=1 prepended by Alois P. Heinz, Jul 22 2024

A100777 Square-factorial numbers: a(1) = 1, a(n+1) = a(n) * largest square divisor of (n+1).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 16, 144, 144, 144, 576, 576, 576, 576, 9216, 9216, 82944, 82944, 331776, 331776, 331776, 331776, 1327104, 33177600, 33177600, 298598400, 1194393600, 1194393600, 1194393600, 1194393600, 19110297600, 19110297600, 19110297600, 19110297600

Offset: 1

Amarnath Murthy, Nov 28 2004

Complementary to A048803 which can be defined as squarefree factorials.

Partial products of A008833. - Ray Chandler, Nov 29 2004

Amiram Eldar, Table of n, a(n) for n = 1..2135

Cf. A008833, A048803.

f[p_, e_] := p^(2*Floor[e/2]); s[n_] := Times @@ (f @@@ FactorInteger[n]); FoldList[Times, 1, Array[s, 31, 2]] (* Amiram Eldar, Dec 07 2020 *)

More terms from Amiram Eldar, Dec 07 2020

A239682 Product_{i=1..n} A173557(i).

Original entry on oeis.org

1, 1, 2, 2, 8, 16, 96, 96, 192, 768, 7680, 15360, 184320, 1105920, 8847360, 8847360, 141557760, 283115520, 5096079360, 20384317440, 244611809280, 2446118092800, 53814598041600, 107629196083200, 430516784332800, 5166201411993600, 10332402823987200

Offset: 1

Tom Edgar, Mar 24 2014

nonn

This is the generalized factorial for A173557.

			The first five terms of A173557 are 1,1,2,1,4 so a(5)=4*1*2*1*1=8.

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. A173557, A085542, A048803.

q=50 # change q for more terms
R=[prod([(x-1) for x in prime_divisors(n)]) for n in [1..q]]
[prod(R[0:i+1]) for i in [0..q-1]]

Product_{i=1..n} A173557(i).

a(n) = abs(A085542(n)).

A276998 Coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x) where B_k(x) are the Bernoulli polynomials.

Original entry on oeis.org

1, 1, 2, 1, 12, 6, -1, 72, 24, -24, 1, 1440, 120, -960, 200, 37, 43200, -9360, -44280, 20640, 3750, -1493, 1814400, -997920, -2484720, 2028600, 271740, -378966, 14017, 25401600, -23042880, -42497280, 54159840, 3328080, -18236064, 1977248, 751267

Offset: 0

Peter Luschny, Oct 03 2016

			Sequence of rational polynomials P_n(x) starts:
1;
1;
(2*x + 1)/2;
(12*x^2 + 6*x - 1)/6;
(72*x^3 + 24*x^2 - 24*x + 1)/12;
(1440*x^4 + 120*x^3 - 960*x^2 + 200*x + 37)/60;
(43200*x^5 - 9360*x^4 - 44280*x^3 + 20640*x^2 + 3750*x - 1493)/360;
Triangle starts:
[1]
[1]
[2, 1]
[12, 6, -1]
[72, 24, -24, 1]
[1440, 120, -960, 200, 37]
[43200, -9360, -44280, 20640, 3750, -1493]

T(n,0) = A277174(n)/n for n>=1.

Cf. A048803, A276996, A276997.

P := proc(n) local B;
B := (n, x) -> CompleteBellB(n, seq(k!*bernoulli(k, x), k=0..n)):
sort(A048803(n)*B(n, x)) end:
A276998_row := n -> PolynomialTools[CoefficientList](P(n), x, termorder=reverse):
seq(op(A276998_row(n)), n=0..8);
# Recurrence for the rational polynomials:
A276998_poly := proc(n,x) option remember; local z; if n = 0 then return 1 fi;
z := proc(k) option remember; k!*bernoulli(k,x) end;
expand(add(binomial(n-1,j)*z(n-j-1)*A276998_poly(j,x),j=0..n-1)) end:
for n from 0 to 5 do sort(A276998_poly(n,x)) od;

(* b = A048803 *) b[0] = 1; b[n_] := b[n] = b[n-1] First @ Select[ Reverse @ Divisors[n], SquareFreeQ, 1];
CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
B[n_, x_] := CompleteBellB[n, Table[k!*BernoulliB[k, x], {k, 0, n}]];
P[n_] := b[n] B[n, x];
row[0] = {1}; row[n_] := CoefficientList[P[n], x] // Reverse;
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Sep 09 2018 *)

P_n(x) = Y_n(x_0, x_1, x_2,..., x_n), the complete Bell polynomials evaluated at x_k = k!*B_k(x) and B_k(x) the Bernoulli polynomials.

T(n,k) = A048803(n)*[x^k] P_n(x).

cp's OEIS Frontend

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

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A368092 a(n) = A160014(m, n) * a(n - 1) for m = 2 and n > 0, a(0) = 1.

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A368093 Cumulative products of the generalized Clausen numbers. Array read by ascending antidiagonals.

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A387126 Triangle read by rows: T(n, k) = (n! / (n - k)!) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

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A387138 Triangle read by rows: T(n, k) = binomial(n, k) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

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A387141 a(n) = floor((Product_{k=1..n} radical(k))^(1/n)) for n >= 1, a(0) = 1, where radical(n) is the product of distinct prime factors of n, cf. A007947.

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A068625 Reduced root factorial of n: product of the smallest integer root of numbers from 1 to n.

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A100777 Square-factorial numbers: a(1) = 1, a(n+1) = a(n) * largest square divisor of (n+1).

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