A002944 -id:A002944 - cp's OEIS frontend

A056531 Sequence remaining after a fourth round of Flavius Josephus sieve; remove every fifth term of A056530.

1, 3, 7, 13, 19, 25, 27, 31, 39, 43, 49, 51, 61, 63, 67, 73, 79, 85, 87, 91, 99, 103, 109, 111, 121, 123, 127, 133, 139, 145, 147, 151, 159, 163, 169, 171, 181, 183, 187, 193, 199, 205, 207, 211, 219, 223, 229, 231, 241, 243, 247, 253, 259, 265, 267, 271, 279

Offset: 1

Henry Bottomley, Jun 19 2000

Numbers {1, 3, 7, 13, 19, 25, 27, 31, 39, 43, 49, 51} mod 60.

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

Compare A000027 for 0 rounds of sieve, A005408 after 1 round of sieve, 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[{1,0,0,0,0,0,0,0,0,0,0,1,-1},{1,3,7,13,19,25,27,31,39,43,49,51,61},60] (* Harvey P. Dale, Mar 11 2019 *)

From Chai Wah Wu, Jul 24 2016: (Start)
a(n) = a(n-1) + a(n-12) - a(n-13) for n > 13.
G.f.: x*(9*x^12 + 2*x^11 + 6*x^10 + 4*x^9 + 8*x^8 + 4*x^7 + 2*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 4*x^2 + 2*x + 1)/(x^13 - x^12 - x + 1). (End)

A099946 a(n) = lcm{1, 2, ..., n}/(n*(n-1)), n >= 2.

Original entry on oeis.org

1, 1, 1, 3, 2, 10, 15, 35, 28, 252, 210, 2310, 1980, 1716, 3003, 45045, 40040, 680680, 612612, 554268, 503880, 10581480, 9699690, 44618574, 41186376, 114406600, 106234700, 2868336900, 2677114440, 77636318760, 145568097675, 136745788725

Offset: 2

N. J. A. Sloane, Nov 12 2004

Alois P. Heinz, Table of n, a(n) for n = 2..1000 (first 99 terms from Luke March)

Cf. A002378, A002944, A003418, A025558.

a:= n-> ilcm(seq(k,k=1..n))/n/(n-1): seq(a(n), n=2..37); # Emeric Deutsch, Jun 13 2005

Table[LCM@@Range[n]/(n(n-1)), {n,2,40}] (* Harvey P. Dale, Jan 14 2011 *)

a(n) = lcm(vector(n, i, i))/(n*(n-1)); \\ Michel Marcus, Jul 25 2014

from math import gcd
def lcm(a, b):
    return (a * b) // gcd(a, b)
def f(lim):
    l = 1
    for n in range(2, lim + 1):
        l = lcm(n, l)
        print(n, l // (n * (n - 1)))
f(100) # Luke March, Jul 23 2014

a(n) = A003418(n)/(n*(n-1)) = A003418(n)/A002378(n-1), n >= 2.

More terms from Emeric Deutsch, Jun 13 2005

A173185 Partial sums of A003418.

Original entry on oeis.org

1, 2, 4, 10, 22, 82, 142, 562, 1402, 3922, 6442, 34162, 61882, 422242, 782602, 1142962, 1863682, 14115922, 26368162, 259160722, 491953282, 724745842, 957538402, 6311767282, 11665996162, 38437140562, 65208284962, 145521718162, 225835151362, 2554924714162

Offset: 0

Jonathan Vos Post, Feb 12 2010

From Antti Karttunen, Feb 27 2014: (Start)
For all n >= 4, a(n) mod 10 = 2 (as A003418(5) = 60, the first multiple of ten in that sequence).
For all n >= 24, a(n) mod 100 = 62 (as A003418(25) = 26771144400, the first multiple of one hundred in that sequence).
Cf. also A236856.
a(n-1) gives the position of the first element of row n in irregular tables like A238280.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2297 (first 251 terms from Antti Karttunen)

One more than A236856.

Cf. A002944, A003418, A003422, A102910, A093880, A133233, A231721, A236857, A236858, A238280.

b:= proc(n) b(n):= `if`(n=0, 1, ilcm(n, b(n-1))) end:
a:= proc(n) a(n):= `if`(n<0, 0, a(n-1) +b(n)) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 31 2018

Table[If[n == 0, 1, LCM @@ Range[n]], {n, 0, 50}] // Accumulate (* Jean-François Alcover, Jan 03 2022 *)

a(n) = sum(k=0, n, lcm(vector(k, i, i))); \\ Michel Marcus, Mar 13 2018

(define (A173185 n) (if (< n 1) 1 (+ (A173185 (- n 1)) (A003418 n))))

a(n) = Sum_{i=0..n} A003418(i).

Missing term a(9)=3922 inserted by Antti Karttunen, Feb 27 2014

A349203 Triangle read by rows, T(n, k) = (lcm_{k=0..n} binomial(n, k)) / binomial(n, k).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 1, 3, 12, 3, 2, 3, 12, 10, 2, 1, 1, 2, 10, 60, 10, 4, 3, 4, 10, 60, 105, 15, 5, 3, 3, 5, 15, 105, 280, 35, 10, 5, 4, 5, 10, 35, 280, 252, 28, 7, 3, 2, 2, 3, 7, 28, 252, 2520, 252, 56, 21, 12, 10, 12, 21, 56, 252, 2520

Offset: 0

Peter Luschny, Nov 13 2021

			Triangle starts:
[0]   1;
[1]   1,  1;
[2]   2,  1,  2;
[3]   3,  1,  1, 3;
[4]  12,  3,  2, 3, 12;
[5]  10,  2,  1, 1,  2, 10;
[6]  60, 10,  4, 3,  4, 10, 60;
[7] 105, 15,  5, 3,  3,  5, 15, 105;
[8] 280, 35, 10, 5,  4,  5, 10,  35, 280;
[9] 252, 28,  7, 3,  2,  2,  3,   7,  28, 252;

Cf. A007318, A347563, A025533 (row sums), A002944 (column 0 and main diagonal).

b := n -> ilcm(seq(binomial(n, k), k=0..n)):
A349203 := (n, k) -> b(n)/binomial(n, k):
seq(seq(A349203(n, k), k = 0..n), n = 0..11);

A363154 Triangle read by rows. The Hadamard product of A173018 and A349203.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 4, 1, 0, 12, 33, 22, 3, 0, 10, 52, 66, 26, 2, 0, 60, 570, 1208, 906, 228, 10, 0, 105, 1800, 5955, 7248, 3573, 600, 15, 0, 280, 8645, 42930, 78095, 62476, 21465, 2470, 35, 0, 252, 14056, 102256, 264702, 312380, 176468, 43824, 3514, 28, 0

Offset: 0

Peter Luschny, May 21 2023

			Triangle T(n, k) starts:
[0]   1;
[1]   1,    0;
[2]   2,    1,     0;
[3]   3,    4,     1,     0;
[4]  12,   33,    22,     3,     0;
[5]  10,   52,    66,    26,     2,     0;
[6]  60,  570,  1208,   906,   228,    10,    0;
[7] 105, 1800,  5955,  7248,  3573,   600,   15,  0;
[8] 280, 8645, 42930, 78095, 62476, 21465, 2470, 35, 0;

Peter Luschny, A combinatorial triangle for the Bernoulli numbers, MathOverflow 2023.

Cf. A173018, A349203, A002944 (column 0), A099946, A362994 (alternating row sums), A362990 (row sums).

A173018 := (n, k) -> combinat[eulerian1](n, k):
A349203 := (n, k) -> ilcm(seq(binomial(n, j), j = 0..n)) / binomial(n, k):
A363154 := (n, k) -> A173018(n, k) * A349203(n, k):
for n from 0 to 8 do seq(A363154(n, k), k = 0..n) od;

T(n, k) = A173018(n, k) * A349203(n, k).

Sum_{k=0..n} (-1)^k * T(n, k) = lcm(1, 2, ..., n+1)*Bernoulli(n, 1) = A362994(n).

A334721 Denominator of the product of n and the n-th harmonic alternating number, Sum_{k=1..n} (-1)^(k+1)/k.

Original entry on oeis.org

1, 1, 2, 3, 12, 10, 60, 105, 280, 252, 2520, 2310, 27720, 25740, 24024, 9009, 144144, 136136, 2450448, 11639628, 11085360, 10581480, 232792560, 223092870, 1070845776, 1029659400, 2974571600, 2868336900, 11473347600, 11090902680, 332727080400, 644658718275, 625123605600

Offset: 1

Petros Hadjicostas, May 08 2020

For n = 1 to 15, we have a(n) = A002944, but a(16) = 9009 <> 45045 = A002944(16).

			The first few fractions are 1, 1, 5/2, 7/3, 47/12, 37/10, 319/60, 533/105, 1879/280, ... = A119787/A334721.

Cf. A002944, A119787 (numerators).

a(n) = denominator(n*sum(k=1, n, (-1)^(k+1)/k)); \\ Michel Marcus, May 09 2020

A362991 Triangle read by rows. T(n, k) = lcm{1, 2, ..., n+1} * Sum_{j=0..n-k} (-1)^(n-k-j) * j! * Stirling2(n - k, j) / (j + k + 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 0, 2, 3, 3, -2, 2, 9, 12, 12, 0, -2, 3, 8, 10, 10, 10, -10, -9, 24, 50, 60, 60, 0, 20, -30, -8, 50, 90, 105, 105, -84, 84, 18, -96, 0, 150, 245, 280, 280, 0, -84, 126, -24, -90, 18, 147, 224, 252, 252, 2100, -2100, 126, 1344, -600, -870, 343, 1568, 2268, 2520, 2520

Offset: 0

Peter Luschny, May 16 2023

A variant of the Akiyama-Tanigawa algorithm for the Bernoulli numbers A164555/ A027642.

			Triangle T(n, k) starts:
[0]   1;
[1]   1,   1;
[2]   1,   2,   2;
[3]   0,   2,   3,   3;
[4]  -2,   2,   9,  12,  12;
[5]   0,  -2,   3,   8,  10,  10;
[6]  10, -10,  -9,  24,  50,  60,  60;
[7]   0,  20, -30,  -8,  50,  90, 105, 105;
[8] -84,  84,  18, -96,   0, 150, 245, 280, 280;
[9]   0, -84, 126, -24, -90,  18, 147, 224, 252, 252;

Paolo Xausa, Table of n, a(n) for n = 0..11324 (rows 0..150 of the triangle, flattened)
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
Index entries for sequences related to Bernoulli numbers.

Variant: A051714/A051715.

Cf. A362994 (column 0), A002944 (main diagonal), A164555/A027642 (Bernoulli).

LCM := n -> ilcm(seq((1 + i), i = 0..n)):
T := (n, k) -> LCM(n)*add((-1)^(n - k - j)*j!*Stirling2(n - k, j)/(j + k + 1), j = 0..n - k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

A362991row[n_]:=Table[LCM@@Range[n+1]Sum[(-1)^(n-k-j)j!StirlingS2[n-k,j]/(j+k+1),{j,0,n-k}],{k,0,n}];Array[A362991row,15,0] (* Paolo Xausa, Aug 09 2023 *)

def A362991Triangle(size):  # 'size' is the number of rows.
    A, T, l = [], [], 1
    for n in range(size):
        A.append(Rational(1/(n + 1)))
        for j in range(n, 0, -1):
            A[j - 1] = j * (A[j - 1] - A[j])
        l = lcm(l, n + 1)
        T.append([a * l for a in A])
    return T
A362991Triangle(10)

T(n, 0) = lcm(1, 2, ..., n+1) * Bernoulli(n, 1).

A025532 a(n) is the sum of exponents in the prime factorization of lcm{C(n,0), C(n,1), ..., C(n,n)}.

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 4, 3, 5, 5, 7, 5, 8, 7, 7, 6, 10, 8, 11, 9, 10, 10, 12, 9, 12, 12, 12, 12, 15, 13, 16, 13, 16, 16, 16, 14, 18, 17, 17, 15, 19, 17, 20, 18, 18, 19, 21, 17, 21, 20, 21, 20, 23, 20, 22, 20, 22, 22, 24, 21, 25, 24, 23, 21, 25, 24, 27, 25, 26, 25, 28, 24, 29, 28, 27

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

{0, 0}~Join~Table[Total@ FactorInteger[LCM @@ Array[Binomial[n, #] &, n]][[All, -1]], {n, 2, 74}] (* Michael De Vlieger, Jan 13 2018 *)

for(n=0, 100, l=1; for(k=0, n, l=lcm(l,binomial(n,k))); v=factor(l); s=0; for(k=1, matsize(v)[1], s=s+v[k,2]); print1(s","))

a(n) = bigomega(lcm(vector(n+1, k, binomial(n, k-1)))); \\ Michel Marcus, Jan 06 2018

a(n) = A025528(n + 1) - A001222(n + 1). - Luc Rousseau, Jan 04 2018

a(n) = A001222(A002944(n+1)). - Michel Marcus, Jan 05 2018

More terms from Ralf Stephan, Mar 28 2003

A056609 a(n) = rad(n!)/rad(A001142(n)) where rad(n) is the squarefree kernel of n, A007947(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 1, 1, 7, 5, 2, 1, 3, 1, 5, 7, 11, 1, 1, 5, 13, 3, 7, 1, 1, 1, 2, 11, 17, 7, 1, 1, 19, 13, 1, 1, 7, 1, 11, 1, 23, 1, 1, 7, 5, 17, 13, 1, 3, 11, 1, 19, 29, 1, 1, 1, 31, 1, 2, 13, 11, 1, 17, 23, 1, 1, 1, 1, 37, 5, 19, 11, 13, 1, 1, 3, 41, 1, 1, 17, 43, 29, 11, 1, 1, 13

Offset: 1

Labos Elemer, Aug 07 2000

nonn

The previous name, which does not match the data as observed by Luc Rousseau, was: Quotient of squarefree kernels of A002944(n) and A001405.

a(n) is the unique prime p not greater than n missing in the prime factorization of A001142(n), if such a prime exists; a(n) is 1 otherwise. - Luc Rousseau, Jan 01 2019

			From _Luc Rousseau_, Jan 02 2019: (Start)
In Pascal's triangle,
- row n=3 (1 3 3 1) contains no number with prime factor 2, so a(3) = 2;
- row n=4 (1 4 6 4 1) contains, for all p prime <= 4, a multiple of p, so a(4) = 1;
- row n=5 (1 5 10 10 5 1) contains no number with prime factor 3, so a(5) = 3;
etc.
(End)

Luc Rousseau, Table of n, a(n) for n = 1..1000 (first 90 terms from Labos Elemer)

Cf. A002944, A001405, A003418, A002110, A001142, A034386, A056606.

L[n_] := Table[Binomial[n, k], {k, 1, Floor[n/2]}]
c[n_] := Complement[Prime /@ Range[PrimePi[n]], First /@ FactorInteger[Times @@ L[n]]]
a[n_] := Module[{x = c[n]}, If[x == {}, 1, First[x]]]
Table[a[n], {n, 1, 100}]
(* Luc Rousseau, Jan 01 2019 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
b(n) = prod(m=1, n, binomial(n, m)); \\ A001142
a(n) = rad(n!)/rad(b(n)); \\ Michel Marcus, Jan 02 2019

a(n) = A034386(n) / A056606(n). - Sean A. Irvine, Apr 24 2022

Definition and example changed by Luc Rousseau, Jan 02 2019

A064451 LCM of totients of binomial coefficients C(n,j), j = 0..n.

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 24, 24, 72, 288, 240, 240, 1440, 2880, 2880, 11520, 23040, 46080, 207360, 276480, 82944, 829440, 2280960, 9123840, 15206400, 60825600, 273715200, 1642291200, 766402560, 7664025600, 1916006400, 1277337600

Offset: 1

Labos Elemer, Oct 02 2001

nonn

			For n=4, the binomial coefficients C(4,j) are 1, 4, 6, 4, and 1. The totients are 1, 2, 2, 2, and 1.  So a(4) = lcm of 1, 2, 2, 2, 1 = 2. - _Michael B. Porter_, Jun 25 2018

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A002944 (see 1st comment there).

Table[LCM@@Table[EulerPhi[Binomial[n,j]],{j,0,n}],{n,40}] (* Harvey P. Dale, Nov 04 2019 *)

a(n) = lcm(vector(n+1, k, eulerphi(binomial(n, k-1)))); \\ Michel Marcus, Jun 24 2018

Previous Mathematica program replaced by Harvey P. Dale, Nov 04 2019

cp's OEIS Frontend

A056531 Sequence remaining after a fourth round of Flavius Josephus sieve; remove every fifth term of A056530.

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A099946 a(n) = lcm{1, 2, ..., n}/(n*(n-1)), n >= 2.

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A173185 Partial sums of A003418.

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A349203 Triangle read by rows, T(n, k) = (lcm_{k=0..n} binomial(n, k)) / binomial(n, k).

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A363154 Triangle read by rows. The Hadamard product of A173018 and A349203.

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A334721 Denominator of the product of n and the n-th harmonic alternating number, Sum_{k=1..n} (-1)^(k+1)/k.

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A362991 Triangle read by rows. T(n, k) = lcm{1, 2, ..., n+1} * Sum_{j=0..n-k} (-1)^(n-k-j) * j! * Stirling2(n - k, j) / (j + k + 1).

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A025532 a(n) is the sum of exponents in the prime factorization of lcm{C(n,0), C(n,1), ..., C(n,n)}.

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