A002944 -id:A002944 - cp's OEIS frontend

A027446 Triangle read by rows: square of the lower triangular mean matrix.

1, 3, 1, 11, 5, 2, 25, 13, 7, 3, 137, 77, 47, 27, 12, 147, 87, 57, 37, 22, 10, 1089, 669, 459, 319, 214, 130, 60, 2283, 1443, 1023, 743, 533, 365, 225, 105, 7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280, 7381, 4861, 3601, 2761, 2131, 1627, 1207, 847, 532, 252

Offset: 1

Olivier Gérard

Numerators of nonzero elements of A^2, written as rows using the least common denominator, where A[i,j] = 1/i if j <= i, 0 if j > i. [Edited by M. F. Hasler, Nov 05 2019]

			Triangle starts
     1
     3,    1
    11,    5,    2
    25,   13,    7,    3
   137,   77,   47,   27,   12
   147,   87,   57,   37,   22,   10
  1089,  669,  459,  319,  214,  130,  60
  2283, 1443, 1023,  743,  533,  365, 225, 105
  7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280
  ... - _Joerg Arndt_, Mar 29 2013

L. Bendersky, Sur la fonction gamma généralisée, Acta Math. 61 (1933), p. 263-322. See p. 295.

The row sums give A081528(n), n>=1.
The column sequences give A025529, A027457, A027458 for j=1..3.
The diagonal sequences give A002944, A027449, A027450.
Cf. A027447, A027448.

rows = 10;
M = MatrixPower[Table[If[j <= i, 1/i, 0], {i, 1, rows}, {j, 1, rows}], 2];
T = Table[M[[n]]*LCM @@ Denominator[M[[n]]], {n, 1, rows}];
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 05 2013, updated May 06 2022 *)

A027446_upto(n)={my(M=matrix(n, n, i, j, (j<=i)/i)^2); vector(n,r,M[r,1..r]*denominator(M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

The rational matrix A^2, where the matrix A has elements a[i,j] = 1/A002024(i,j), is equal to A119947(i,j)/A119948(i,j).

a(i,j) = lcm(seq(A119948(i,m),m=1..i))*A119947(i,j)/A119948(i,j), 1 <= j =< i and zero otherwise.

Edited by M. F. Hasler, Nov 05 2019

A056530 Sequence remaining after third round of Flavius Josephus sieve; remove every fourth term of A047241.

Original entry on oeis.org

1, 3, 7, 13, 15, 19, 25, 27, 31, 37, 39, 43, 49, 51, 55, 61, 63, 67, 73, 75, 79, 85, 87, 91, 97, 99, 103, 109, 111, 115, 121, 123, 127, 133, 135, 139, 145, 147, 151, 157, 159, 163, 169, 171, 175, 181, 183, 187, 193, 195, 199, 205, 207, 211, 217, 219, 223, 229, 231

Offset: 1

Henry Bottomley, Jun 19 2000

Numbers {1, 3, 7} mod 12: A017533, A017557, A017605 interleaved.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for sequences related to the Josephus Problem

We have A000027 after 0 rounds of sieving, A005408 after 1 round of sieving, A047241 after 2 rounds, A056530 after 3 rounds, A056531 after 4 rounds, A000960 after all rounds. After n rounds the remaining sequence comprises A002944(n) numbers mod A003418(n+1), i.e. 1/(n+1) of them.

LinearRecurrence[{1,0,1,-1},{1,3,7,13},60] (* Harvey P. Dale, Oct 19 2022 *)

From Chai Wah Wu, Jul 24 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
G.f.: x*(5*x^3 + 4*x^2 + 2*x + 1)/(x^4 - x^3 - x + 1). (End)
a(n) = 4*n - (13 + 2*A131713(n))/3. - R. J. Mathar, Jun 22 2020

A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 30, 105, 70, 42, 210, 2310, 2310, 4290, 6006, 15015, 30030, 170170, 510510, 1939938, 1385670, 881790, 9699690, 223092870, 44618574, 17160990, 74364290, 31870410, 223092870, 6469693230, 6469693230, 100280245065

Offset: 0

Labos Elemer, Aug 07 2000

nonn

Also squarefree kernel of A001142; row products in table A256113. - Reinhard Zumkeller, Mar 21 2015
a(2372) has 1001 decimal digits. - Michael De Vlieger, Jul 14 2017
Also the squarefree kernel of the cumulative product of n^n/n!. - Peter Luschny, Dec 21 2019
Conjecture: the few odd values belong to A070826. - Bill McEachen, Jun 24 2023
And their indices appear to be A007053. - Michel Marcus, Jul 01 2023

			a(7) = 105 because lcm(1, 7, 21, 35) = 105 is already squarefree.
a(0) = 1 because n^n/n! = 1 for the integer n = 0. - _Peter Luschny_, Dec 21 2019

Michael De Vlieger, Table of n, a(n) for n = 0..2371

Cf. A002944, A034386, A048633, A003418, A000142, A001405.
Cf. A007947, A001142, A256113, A002109, A000178, A329947.
Cf. A070826.

a056606 = a007947 . a001142  -- Reinhard Zumkeller, Mar 21 2015

h := n -> mul(k^k/factorial(k), k=0..n):
rad := n -> mul(k, k = numtheory[factorset](n)):
seq(rad(h(n)), n=0..31); # Peter Luschny, Dec 21 2019

Table[Apply[Times, FactorInteger[Product[k^(2 k - 1 - n), {k, n}]][[All, 1]]], {n, 0, 31}] (* or *)
Table[Apply[Times, FactorInteger[Apply[LCM, Range@ n]/n][[All, 1]]], {n, 1, 32}] (* Michael De Vlieger, Jul 14 2017 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = rad(lcm(vector(n+1, k, binomial(n,k-1)))); \\ Michel Marcus, Jun 24 2023

a(n) = A007947(A002944(n+1)). - Michel Marcus, Dec 21 2019

a(n) = radical(hyperfactorial(n)/superfactorial(n)) = A007947(A002109(n)/ A000178(n)) for n >= 0. - Peter Luschny, Dec 21 2019

Extended with a(0) = 1 by Peter Luschny, Dec 21 2019

A339453 Number of subsets of {1..n} whose harmonic mean is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 13, 14, 15, 18, 19, 26, 27, 30, 53, 54, 55, 100, 101, 180, 203, 210, 211, 378, 379, 382, 383, 1092, 1093, 2020, 2021, 2022, 3933, 3956, 6473, 10226, 10227, 10266, 10561, 20948, 20949

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

For terms listed in the Data section: a(p^k) = a(p^k-1) + 1, where p prime (empirical observation). - Ilya Gutkovskiy, Dec 06 2020
From Chai Wah Wu, Dec 14 2020: (Start)
The above empirical observation is true.
Theorem: For prime p, a(p^k) = a(p^k-1)+1.
Proof: Since the singleton set {x} has harmonic mean x, a(n) >= a(n-1)+1.
Let S = {s_1,s_2,..,s_n} be a subset of {1,2,..,p^k} with n>1 elements such that s_n = p^k and let H be the harmonic mean of S. Let M = A003418(p^k) be the least common multiple of {1,2,..,p^k}. Then M = Wp^k where p does not divide W = A002944(p^k).
Let Q_i = M/s_i and Q = sum_i Q_i. This implies that Q_n = W and p divides Q_i for i < n.
H can be written as nM/Q. Since p does not divide W, this implies that p does not divide Q. Suppose H is an integer. Then this implies that Q divides nM/p^k = nW.
Note that s_i < s_n for i < n. This implies that Q_i > W for i < n, i.e. Q > nW, and this contradicts the fact that Q divides nW and thus H is not an integer.
Thus {p^k} is the only subset of {1,2,..,p^k} that includes p^k and have an integral Harmonic mean.
This concludes the proof.
(End)

			a(6) = 12 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {2, 6}, {3, 6}, {1, 3, 6}, {2, 3, 6}, {3, 4, 6} and {1, 2, 3, 6}.

Eric W. Weisstein's World of Mathematics, Harmonic Mean

Cf. A002944, A003418, A051293, A067540, A246655, A326027, A339454.

from itertools import chain, combinations
from fractions import Fraction
def powerset(s): # skip empty set
    return chain.from_iterable(combinations(s, r) for r in range(1,len(s)+1))
def hm(s):
    ss = sum(Fraction(1, i) for i in s)
    return Fraction(len(s)*ss.denominator, ss.numerator)
def a(n):
    return sum(hm(s).denominator==1 for s in powerset(range(1,n+1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Dec 06 2020

from math import lcm
from itertools import combinations
def A339453(n):
    m = lcm(*range(2,n+1))
    return sum(1 for i in range(1,n+1) for d in combinations((m//i for i in range(1,n+1)),i) if m*i % sum(d) == 0) # Chai Wah Wu, Dec 02 2021

a(n) >= a(n-1)+1. For prime p, a(p^k) = a(p^k-1)+1. - Chai Wah Wu, Dec 14 2020

a(n) = A357411(n) + A357412(n). - Max Alekseyev, Feb 26 2025

a(23)-a(29) from Michael S. Branicky, Dec 06 2020
a(30)-a(35) from Chai Wah Wu, Dec 08 2020
a(36)-a(39) from Chai Wah Wu, Dec 11 2020
a(40)-a(41) from Chai Wah Wu, Dec 19 2020

A004788 Number of distinct prime divisors of the numbers in row n of Pascal's triangle.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 14, 13, 14, 15, 14, 14, 14, 14, 15, 15, 15, 16, 15, 15, 16, 17, 17, 17, 18, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20

Offset: 0

Clark Kimberling

nonn

Also the number of prime divisors of A002944(n) = lcm_{j=0..floor(n/2)} binomial(n,j).
The terms are increasing by intervals, then decrease once. The local maxima are obtained for 23, 44, 47, 55, 62, 79, 83, 89, 104, 119, 131, 134, 139, 143, .... - Michel Marcus, Mar 21 2013
a(A004789(n)) = n and a(m) != n for m < A004789(n). - Reinhard Zumkeller, Mar 16 2015

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A004789.

Cf. A001221, A001142, A256113.

a004788 = a001221 . a001142  -- Reinhard Zumkeller, Mar 16 2015

Table[prd = Product[Binomial[n, k], {k, 0, n}]; If[prd == 1, 0, Length[FactorInteger[prd]]], {n, 0, 100}] (* T. D. Noe, Mar 21 2013 *)

a(n) = {sfp = Set(); for (k=1, n-1, sfp = setunion(sfp, Set(factor(binomial(n, k))[,1]))); return (length(sfp));} \\ Michel Marcus, Mar 21 2013

a(n) = A001221(A001142(n)). - Reinhard Zumkeller, Mar 16 2015

A190940 Number of divisors of LCM(1,2,...,n)/n.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 12, 8, 16, 18, 48, 32, 96, 72, 64, 48, 240, 128, 480, 288, 320, 384, 960, 512, 960, 1152, 960, 1152, 3840, 3072, 7680, 3072, 6912, 7680, 6144, 6144, 18432, 15360, 13824, 12288, 36864, 23040, 73728, 49152, 49152, 61440, 147456, 73728, 147456, 122880

Offset: 1

Naohiro Nomoto, May 24 2011

nonn

Also, number of sequences of d1 = 1 < d2 < ... < dk = n for some k >= 1 that are the first k divisors of some integer (cf. A378314). - Max Alekseyev, Nov 22 2024
Also, the number of distinct values taken by lcm(a,a+b,a+b+c,...,n), where positive integers a,b,c,... run over the compositions a+b+c+...=n. - Conjectured by Ridouane Oudra, Aug 24 2019; proved by Max Alekseyev, Nov 22 2024
Proof. It is clear that n | lcm(a,a+b,...,n) | lcm(1,2,...,n). Hence, lcm(a,a+b,...,n) = d*n for some d | lcm(1,2,...,n)/n. We'll show that each such d is achievable. Suppose d*n has prime factorization p1^e1 * ... * pk^ek with p1^e1 < ... < pk^ek. It is clear that pk^ek <= n, and we can take a composition (a,b,c,...) = (p1^e1, p2^e2 - p1^e1, p3^e3 - p2^e2, ..., pk^ek - p(k-1)^e(k-1), n - pk^ek), which delivers lcm(a,a+b,a+b+c,...,n) = p1^e1 * ... * pk^ek = d*n. QED - Max Alekseyev, Nov 22 2024

			Examples: for n=3 the a(3) = 2 distinct values are 3, 6. The compositions are 3, 1+2, 2+1, and 1+1+1. The values of the lcm are lcm(3)=3, lcm(1,1+2)=3, lcm(2,2+1)=6, and lcm(1,1+1,1+1+1)=6.

First difference of A378314.

Cf. A000005, A002944, A003418, A056793, A101207.

Lpsum := proc(L) local ps,k ; ps := [op(1,L)] ; for i from 2 to nops(L) do ps := [op(ps), op(-1,ps)+op(i,L)] ; end do: ps ; end proc:
A190940 := proc(n) local lc,k,c ; lc := {} ; for k from 1 to n do for c in combinat[composition](n,k) do lc := lc union { ilcm( op(Lpsum(c))) }; end do: end do: nops(lc) ; end proc: # R. J. Mathar, Jun 02 2011

a[n_] := LCM @@@ (Accumulate /@ (Permutations /@ Rest[IntegerPartitions[n]] // Flatten[#, 1]&)) // Union // Length; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Feb 27 2014 *)

a(n) = A000005(A002944(n)).

a(12)-a(20) from R. J. Mathar, Jun 02 2011
a(21)-a(24) from Alois P. Heinz, Nov 03 2011
Edited and terms a(25) onward added by Max Alekseyev, Nov 22 2024

A025558 a(n) = (n/(n+1)) * lcm(1,2,...,n+1).

Original entry on oeis.org

1, 4, 9, 48, 50, 360, 735, 2240, 2268, 25200, 25410, 332640, 334620, 336336, 675675, 11531520, 11571560, 220540320, 221152932, 221707200, 222211080, 5121436320, 5131136010, 25700298624, 25741485000, 77338861600, 77445096300, 2248776129600, 2251453244040

Offset: 1

Clark Kimberling

nonn

a(n) = (1/1 + 1/3 + 1/6 + ... + 1/C(n+1,2))*lcm(1,3,6,...,binomial(n+1,2)) = 2n/(n+1) * lcm(1,3,6,...,binomial(n+1,2)).

a(n+1) = a(n) * ((n+1)^2)/(n * ((n+2)/p) ), where p = n+2 if n+2 is prime, p = q if n+2 = q^k (q is prime, k>1), or p = 1 if n+2 is not a prime or a prime power. - Scott C. Macfarlan (scottmacfarlan(AT)covance.com), Jan 08 2004

Cf. A002944, A003418.

a:= n-> (n/(n+1)) * ilcm($1..n+1):
seq(a(n), n=1..29);  # Alois P. Heinz, Mar 07 2022

Table[n/(n+1) LCM@@Range[n+1],{n,30}]  (* Harvey P. Dale, Apr 02 2011 *)

a(n) = n*lcm([1..n+1])/(n+1); \\ Michel Marcus, Mar 07 2022

a(n) = n * A002944(n+1) = (n/(n+1)) * A003418(n+1).

Entry revised by N. J. A. Sloane, Nov 12 2004

A081528 a(n) = n*lcm{1,2,...,n}.

Original entry on oeis.org

1, 4, 18, 48, 300, 360, 2940, 6720, 22680, 25200, 304920, 332640, 4684680, 5045040, 5405400, 11531520, 208288080, 220540320, 4423058640, 4655851200, 4888643760, 5121436320, 123147264240, 128501493120, 669278610000, 696049754400

Offset: 1

Amarnath Murthy, Mar 27 2003

Denominators in binomial transform of 1/(n + 1)^2. - Paul Barry, Aug 06 2004
Construct a sequence S_n from n sequences b_1, b_2, ..., b_n of periods 1, 2, ..., n, respectively, say, b_1 = [1, 1, ...], b_2 = [1, 2, 1, 2, ...], ..., b_n = [1, 2, 3, ..., n, 1, 2, 3, ..., n, ...], by taking S_n = [b_1(1), b_2(1), ..., b_n(1), b_1(2), b_2(2), ..., b_n(2), ..., b_1(n), b_2(n), ..., b_n(n), ...] (by listing the b_i sequences in rows and taking each column in turn as the next n terms of S_n). Then a(n) is the period of sequence S_n. - Rick L. Shepherd, Aug 21 2006
This is a sequence that goes in strictly ascending order. The related sequence A003418 also goes in ascending order but has consecutive repeated terms. Since n increases, then so too does a(n) even when A003418(n) doesn't. - Alonso del Arte, Nov 25 2012

			a(2) = 4 because the least common multiple of 1 and 2 is 2, and 2 * 2 = 4.
a(3) = 18 because lcm(1,2,3) = 6, and 3 * 6 = 18.
a(4) = 48 because lcm(1, 2, 3, 4) = 12, and 4 * 12 = 48.

Cf. A027612, A027611, A022819, A002944, A081530, A097344.

a(n) := (n + 1)*LCM(VECTOR(k + 1, k, 0, n)) " Paul Barry, Aug 06 2004 "

Table[n*LCM@@Range[n], {n, 30}] (* Harvey P. Dale, Oct 09 2012 *)

l=vector(35); l[1]=1; print1("1, "); for(n=2,35, l[n]=lcm(l[n-1],n); print1(n*l[n],", ")) \\ Rick L. Shepherd, Aug 21 2006

a(n) = A003418(n) * n. - Martin Fuller, Jan 03 2006

More terms from Paul Barry, Aug 06 2004

Entry revised by N. J. A. Sloane, Jan 15 2006

A027451 First diagonal of A027447.

Original entry on oeis.org

1, 1, 4, 9, 144, 100, 3600, 11025, 78400, 63504, 6350400, 5336100, 768398400, 662547600, 577152576, 2029052025, 519437318400, 463325262400, 150117385017600, 135480939978384, 122885206329600, 111967718990400, 54192375991353600, 49770428644836900

Offset: 1

Olivier Gérard

nonn

Equals the denominators of MN(z;n)/(n!)^2 for n =>1, see A162990. - Johannes W. Meijer, Jul 21 2009

It appears that a(n) = denominator of n^2*sum(1/k^2,k=1..n). - Gary Detlefs, May 29 2010

From Johannes W. Meijer, Jul 21 2009: (Start)
Equals A002944(n)^2.
Equals A001044(n-1)/A025527(n)^2.
(End)

a[n_] := (LCM @@ Range[n]/n)^2; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 05 2013, after Johannes W. Meijer *)

Numerators of sequence a[ n, n ] in (a[ i, j ])^3 where a[ i, j ] = 1/i if j<=i, 0 if j>i.

a(n) = (lcm($1..n)/n)^2. - Johannes W. Meijer, Jul 21 2009

More terms from Sean A. Irvine, Nov 04 2019

A048619 a(n) = LCM(binomial(n,0), ..., binomial(n,n)) / binomial(n,floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 3, 4, 2, 10, 5, 30, 15, 7, 7, 56, 28, 252, 126, 60, 30, 330, 165, 396, 198, 286, 143, 2002, 1001, 15015, 15015, 7280, 3640, 1768, 884, 15912, 7956, 3876, 1938, 38760, 19380, 406980, 203490, 99484, 49742, 1144066, 572033, 1961256, 980628

Offset: 0

Labos Elemer

			If n=10 then A002944(10)=2520, A001405(10)=252, the quotient a(10)=10.

Index entries for sequences related to lcm's

Cf. A001405, A002944.

[Lcm([1..n+1]) div (Floor((n+3)/2)*Binomial(n+1,Floor((n+3)/2))): n in [0..50]]; // Vincenzo Librandi, Jul 10 2019

Table[Apply[LCM, Binomial[n, Range[0, n]]]/Binomial[n, Floor[n/2]], {n, 0, 48}] (* Michael De Vlieger, Jun 29 2017 *)

{A048619(n) = lcm(vector(n+1, i, i)) / binomial(n+1, (n+1)\2) / ((n+2)\2);}

a(n) = A002944(n)/A001405(n).
a(n) = lcm(1..n+1)/(floor((n+3)/2)*binomial(n+1,floor((n+3)/2))). - Paul Barry, Jul 03 2006
a(n) = lcm(1,2,...,n+1) / (ceiling((n+1)/2)*binomial(n+1,floor((n+1)/2))) = A003418(n+1) / A100071(n+1). - Max Alekseyev, Oct 23 2015
a(n) = A263673(n+1) / A110654(n+1) = A180000(n+1) / A152271(n). - Max Alekseyev, Oct 23 2015
a(2*n-1) = A068553(n) = A068550(n)/n.

Definition corrected and a(0)=1 prepended by Max Alekseyev, Oct 23 2015

cp's OEIS Frontend

A027446 Triangle read by rows: square of the lower triangular mean matrix.

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A056530 Sequence remaining after third round of Flavius Josephus sieve; remove every fourth term of A047241.

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A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

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A339453 Number of subsets of {1..n} whose harmonic mean is an integer.

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A004788 Number of distinct prime divisors of the numbers in row n of Pascal's triangle.

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A190940 Number of divisors of LCM(1,2,...,n)/n.

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A025558 a(n) = (n/(n+1)) * lcm(1,2,...,n+1).

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