A090624 -id:A090624 - cp's OEIS frontend

A091136 Smallest number m such that number of times m divides k! is almost k/n for large k, i.e., smallest m with A090624(m)=n.

1, 2, 3, 8, 5, 32, 7, 128, 25, 512, 11, 2048, 13, 8192, 2187, 32768, 17, 131072, 19, 524288, 121, 2097152, 23, 8388608, 169, 33554432, 1594323, 134217728, 29, 536870912, 31, 2147483648, 289, 8589934592, 129140163, 34359738368, 37

Offset: 0

Henry Bottomley, Dec 19 2003

nonn

			a(2)=3 noting that 100! is a multiple of 3^48 and 48 is almost 100/2.

Cf. A090622, A090624, A091137.

a(n) = min_p{p prime and n divisible by p-1} p^(n/(p-1)).
a(p-1) = p.
a(2n+1) = 2^(2n+1).
Smallest divisor of A091137(n) which is not a divisor of A091137(n-1).

A091137 The Hirzebruch numbers. a(n) = Product_{2 <= p <= n+1, p prime} p^floor(n / (p - 1)).

Original entry on oeis.org

1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 479001600, 958003200, 2615348736000, 5230697472000, 31384184832000, 62768369664000, 32011868528640000, 64023737057280000, 51090942171709440000, 102181884343418880000, 33720021833328230400000, 67440043666656460800000

Offset: 0

Henry Bottomley, Dec 19 2003

nonn

Largest number m such that number of times m divides k! is almost k/n for large k, i.e., largest m with A090624(m) = n.
This is always a relatively small multiple of n!, since the multiplicity with which a prime p divides n! is always <= n/(p-1); it is equal to floor(n/(p-1)) at least when n is a power of p. - Franklin T. Adams-Watters, May 31 2010
At least for most small n, a(n) = A002790(n) * n!; the first difference is n=15. It appears that A002790(n) * n! always divides a(n).
Conjecture: The denominators of the series reversion of the sequence with e.g.f. Polylog(2,x). - Benedict W. J. Irwin, Jan 05 2017
Not only is a(n) divisible by n!; a(n) is divisible by (n + 1)! as has been observed by Bedhouche and Bakir (see links and A363596). - Hal M. Switkay, Aug 15 2025

			Let n = 4. The partitions of 4 are [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. Thus a(4) = lcm([5, 4*2, 3*3, 3*2*2, 2*2*2*2]) = 720.

P. Curtz, Integration numérique ..., Note 12, C.C.S.A., Arcueil, 1969; see pp. 36, 56.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 3rd. ed., 1966; Lemma 1.7.3, p. 14. [From N. J. A. Sloane, Sep 06 2010]

Michael De Vlieger, Table of n, a(n) for n = 0..443
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See sigma_n p. 3.
Victor M. Buchstaber and Alexander P. Veselov, Todd polynomials and Hirzebruch numbers, arXiv:2310.07383 [math.AT], Oct. 2023.
Donghyun Kim and Jaeseong Oh, Extending the science fiction and the Loehr--Warrington formula, arXiv:2409.01041 [math.CO], 2024. See p. 32.

Starts similarly to A002207 especially for even n and all values of A002207 seen so far seem to divide a(n).

Cf. A002790, A000142, A090622, A090624, A091136, A171080, A238963, A275314, A368093, A368116, A363596.

A091137 := proc(n) local a,i,p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end:
seq(A091137(n), n = 0..47); # R. J. Mathar, Feb 23 2009

A027760[n_] := Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]; a[n_] := a[n] = A027760[n]*a[n-1]; a[0] = 1; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 04 2011 *)

a(n) = local(r); r=1; forprime(p=2, n+1, r*=p^(n\(p-1))); r
\\ Franklin T. Adams-Watters, May 31 2010

from math import prod
from sympy import primerange
def A091137(n): return prod(p**(n//(p-1)) for p in primerange(n+2))
# Chai Wah Wu, Apr 28 2023

def a(n): return lcm(product(r + 1 for r in p) for p in Partitions(n))
# Or, more efficient:
from functools import cache
@cache
def a_rec(n):
    if n == 0: return 1
    p = mul(s for s in map(lambda i: i + 1, divisors(n)) if is_prime(s))
    return p * a_rec(n - 1)
print([a_rec(n) for n in range(22)]) # Peter Luschny, Dec 12 2023

a(n) = Product_p {p prime} p^floor(n/(p-1)).
a(2n+1) = 2*a(2n).
a(n+1) = A027760(n+1)*a(n). - Paul Curtz, Aug 01 2008
From Peter Luschny, Dec 11 2023: (Start)
a(n) = lcm_{p in P(n)} Product_{r in p}(r + 1), where P(n) are the partitions of n.
a(n) = lcm(A238963row(n)).
a(n) = A368116(1, n), seen as the lcm of the product of the 1-shifted partitions.
a(n) = A368093(1, n), seen as the cumulative product of the Clausen numbers A160014(1, n).  (End)
a(n) = lcm({k: A275314(k) = n+1}). - Hal M. Switkay, Aug 13 2025
a(n) = (n + 1)! * A363596(n). - Hal M. Switkay, Aug 15 2025

New name using a formula of the author by Peter Luschny, Dec 11 2023

A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 1, 3, 0, 0, 1, 1, 4, 0, 1, 0, 1, 2, 4, 0, 0, 1, 1, 2, 2, 7, 0, 0, 0, 1, 1, 2, 2, 7, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 0, 1, 1, 2, 1, 4, 4, 10, 0, 0, 0, 1, 1, 1, 1, 4, 2, 4, 5, 10, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11

Offset: 2

Henry Bottomley, Dec 06 2003

			Square array starts:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 0, 0, 1, 0, 0, ...
3, 1, 1, 0, 1, 0, 1, ...
3, 1, 1, 1, 1, 0, 1, ...
4, 2, 2, 1, 2, 0, 1, ...
4, 2, 2, 1, 2, 1, 1, ...
7, 2, 3, 1, 2, 1, 2, ...

Alois P. Heinz, Antidiagonals n = 2..142, flattened

Columns include A011371, A054861, A090616, A027868, A054861, A054896, A090617, A090618, A027868, A064458, A090619, A090620, A054896, A027868, A090621.
Cf. A090623, A090624, A115627.
Diagonal gives A011776. - Alois P. Heinz, Oct 04 2012

f:= proc(n, p) local c, k; c, k:= 0, p;
       while n>=k do c:= c+iquo(n, k); k:= k*p od; c
    end:
T:= (n, k)-> min(seq(iquo(f(n, i[1]), i[2]), i=ifactors(k)[2])):
seq(seq(T(n, 2+d-n), n=2..d), d=2..20);  # Alois P. Heinz, Oct 04 2012

f[n_, p_] := Module[{c = 0, k = p}, While[n >= k , c = c + Quotient[n, k]; k = k*p ]; c ]; t[n_, k_] := Min[ Table[ Quotient[f[n, i[[1]]], i[[2]]], {i, FactorInteger[k]}]]; Table[ Table[t[n, 2 + d - n], {n, 2, d}], {d, 2, 20}] // Flatten (* Jean-François Alcover, Oct 03 2013, translated from Alois P. Heinz's Maple program *)

For k=p prime: T(n,p) = [n/p] + [n/p^2] + [n/p^3] + .... For k = p^m a prime power: T(n,p^m) = [T(n,p)/m]. For k = b*c with b and c coprime: T(n,a*b) = min(T(n,a), T(n,b)). T(n,k) is close to, but below, n/A090624(k).

A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 4, 12, 6, 8, 4, 16, 4, 18, 8, 12, 10, 22, 6, 8, 12, 6, 12, 28, 8, 30, 5, 20, 16, 24, 8, 36, 18, 24, 12, 40, 12, 42, 20, 16, 22, 46, 8, 12, 8, 32, 24, 52, 6, 40, 18, 36, 28, 58, 16, 60, 30, 24, 6, 48, 20, 66, 32, 44, 24, 70, 12, 72, 36, 16

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(60) = a(2^2*3*5) = (2 - 1)*2 * (3 - 1)*1 * (5 - 1)*1 = 16.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A000026, A000040, A002110, A003958, A005117, A005361, A005867, A006093, A007947, A023900, A048250, A059975, A068997, A081294 (numbers n such that a(n) is odd), A087656, A090624, A152649, A173557, A304117, A304408, A304409, A304411, A304412.

Cf. A001221, A076479.

seq(mul((p-1)*padic[ordp](n, p), p in numtheory[factorset](n)), n=1..100); # Ridouane Oudra, Jun 06 2025

a[n_] := Times @@ ((#[[1]] - 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 75}]
Table[EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*abs(A023900(n)) = A005361(n)*A173557(n) = A005361(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*k where p is a prime and k > 0.
a(n) = phi(n) if n is a squarefree (A005117), where phi() = A000010.
a(A002110(k)) = A005867(k).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 1/p^4 - 1/p^5) = 0.2644703894... . - Amiram Eldar, Nov 30 2022
a(n) = (-1)^A001221(n) * (Sum_{d1|n} Sum_{d2|n} mu(d1)*gcd(d1,d2)). - Ridouane Oudra, Jun 06 2025

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 1, 2, 12, 24, 1, 2, 12, 24, 720, 1, 2, 12, 24, 720, 1440, 1, 2, 12, 24, 720, 1440, 60480, 1, 2, 12, 24, 720, 1440, 60480, 120960, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 1, 2, 12

Offset: 1

Paul Curtz, Sep 14 2009

From a study of modified initialization formulas in Adams-Bashforth (1855-1883) multisteps method for numerical integration. On p.36, a(i,j) comes from (j!)*a(i,j) = Integral_{u=i,..,i+1} u*(u-1)*...*(u-j+1) du; see p.32.
Then, with i vertical, j horizontal, with unreduced fractions, partial array is:
0) 1,  1/2,  -1/12,   1/24,  -19/720,   27/1440, ... =  1/log(2)
1) 1,  3/2,   5/12,  -1/24,   11/720,  -11/1440, ... =  2/log(2)
2) 1,  5/2,  23/12,   9/24,  -19/720,   11/1440, ... =  4/log(2)
3) 1,  7/2,  53/12,  55/24,  251/720,  -27/1440, ... =  8/log(2)
4) 1,  9/2,  95/12, 161/24, 1901/720,  475/1440, ... = 16/log(2)
5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... = 32/log(2)
... [improved by Paul Curtz, Jul 13 2019]
First line: the reduced terms are A002206/A002207, logarithmic or Gregory numbers G(n). The difference between the second line and the first one is 0 together A002206/A002207. This is valid for the next lines. - Paul Curtz, Jul 13 2019
See A141417, A140825, A157982, horizontal numerators: A141047, vertical numerators: A000012, A005408, A140811, A141530, A157411. On p.56, coefficients are s(i,q) = (1/q!)* Integral_{u=-i-1,..,1} u*(u+1)*...*(u+q-1) du.
Unreduced fractions array is:
-1) 1,   1/2,  5/12,   9/24, 251/720, 475/1440, ... = A002657/A091137
0)  2,   0/2,  4/12,   8/24, 232/720, 448/1440, ... = A195287/A091137
1)  3,  -3/2,  9/12,   9/24, 243/720, 459/1440, ...
2)  4,  -8/2, 32/12,   0/24, 224/720, 448/1440, ...
3)  5, -15/2, 85/12, -55/24, 475/720, 475/1440, ...
...
(on p.56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281.
From Paul Curtz, Jul 14 2019: (Start)
Difference table from the second line and the first one difference:
1,                -1/2,      -1/12,    -1/24,    -19/720, -27/1440, ...
-3/2,             5/12,       1/24,    11/720,    11/1440, ...
23/12,           -9/24,    -19/720,  -11/1440, ...
-55/24,        251/720,    27/1440, ...
1901/720,    -475/1440,
-4277/1440, ...
...
Compare the lines to those of the first array.
The verticals are the signed diagonals of the first array. (End)

			1;
1,2;
1,2,12;
1,2,12,24;
1,2,12,24,720;

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.

Cf. A090624, A091137.
Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.
Cf. A002206, A002207.

(* a = A091137 *) a[n_] := a[n] = Product[d, {d, Select[Divisors[n]+1, PrimeQ]}]*a[n-1]; a[0]=1; Table[Table[a[k-1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 18 2014 *)

cp's OEIS Frontend

A091136 Smallest number m such that number of times m divides k! is almost k/n for large k, i.e., smallest m with A090624(m)=n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A091137 The Hirzebruch numbers. a(n) = Product_{2 <= p <= n+1, p prime} p^floor(n / (p - 1)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

SageMath

Formula

Extensions

A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica