A022819 -id:A022819 - cp's OEIS frontend

A081527 Erroneous version of A022819.

1, 2, 4, 6, 8, 11, 13, 16, 19, 22, 25, 28, 31, 34, 38, 41, 44, 48, 51, 55, 59, 62, 66, 70, 74, 78, 81, 85, 89, 93, 97, 101, 106, 110, 114, 118, 122, 126, 131, 135, 139, 144, 148, 152, 157, 161, 166, 170, 174, 179, 183, 188, 193, 197, 202, 206, 211, 216

Offset: 1

dead

Corrected and extended by Martin Fuller, Jan 03 2006

A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.

Original entry on oeis.org

1, 5, 13, 77, 87, 223, 481, 4609, 4861, 55991, 58301, 785633, 811373, 835397, 1715839, 29889983, 30570663, 197698279, 201578155, 41054655, 13920029, 325333835, 990874363, 25128807667, 25472027467, 232222818803, 235091155703, 6897956948587, 6975593267347

Offset: 1

Glen Burch (gburch(AT)erols.com)

Numerator of a second-order harmonic number H(n, (2)) = Sum_{k=1..n} HarmonicNumber(k). - Alexander Adamchuk, Apr 12 2006
p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 06 2006
Denominator is A027611(n+1). p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 26 2006
a(n) = A213998(n,n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A001705, A002805, A006675, A022819, A027611, A093418.

import Data.Ratio ((%), numerator)
a027612 n = numerator $ sum $ zipWith (%) [1 .. n] [n, n-1 .. 1]
-- Reinhard Zumkeller, Jul 03 2012

[Numerator((&+[j/(n-j+1): j in [1..n]])): n in [1..30]]; // G. C. Greubel, Aug 23 2022

a := n -> numer(add((n+1-j)/j, j=1..n));
seq(a(n), n = 1..29); # Peter Luschny, May 12 2023

Numerator[Table[Sum[Sum[1/i,{i,1,k}],{k,1,n}],{n,1,30}]] (* Alexander Adamchuk, Apr 12 2006 *)
Numerator[Table[Sum[k/(n-k+1),{k,1,n}],{n,1,50}]] (* Alexander Adamchuk, Jul 26 2006 *)

a(n) = numerator(sum(k=1, n, k/(n-k+1))); \\ Michel Marcus, Jul 14 2018

[numerator(n*(harmonic_number(n+1) - 1)) for n in (1..30)] # G. C. Greubel, Aug 23 2022

From Vladeta Jovovic, Sep 02 2002: (Start)
a(n) = numerators of coefficients in expansion of -log(1-x)/(1-x)^2.
a(n) = numerators of (n+1)*(harmonic(n+1) - 1).
a(n) = numerators of (n+1)*(Psi(n+2) + Euler-gamma - 1). (End)
a(n) = numerator( Sum_{k=1..n} Sum_{i=1..k} 1/i ). - Alexander Adamchuk, Apr 12 2006
a(n) = numerator( Sum_{k=1..n} k/(n-k+1) ). - Alexander Adamchuk, Jul 26 2006
a(n) = numerator of integral_{x=1..n+1} floor((n+1)/x). - Jean-François Alcover, Jun 18 2013

A214075 Triangle read by rows: T(n,k) = floor(A213998(n,k) / A213999(n,k)), 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 2, 0, 1, 4, 7, 6, 2, 0, 1, 5, 12, 14, 8, 2, 0, 1, 6, 17, 26, 22, 11, 2, 0, 1, 7, 24, 44, 49, 34, 13, 2, 0, 1, 8, 31, 68, 93, 83, 47, 16, 2, 0, 1, 9, 40, 100, 162, 177, 131, 64, 19, 2, 0, 1, 10, 49, 140, 263, 340, 309

Offset: 0

Reinhard Zumkeller, Jul 03 2012

T(n,0) = 1;
T(n,1) = n - 1 for n > 0, cf. A001477;
T(n,2) = A074148(n-2) for n > 2;
T(n,n-2) = A022819(n) for n > 1;
T(n,n-1) = A055980(n) for n > 0;
T(n,n) = A000007(n).

			Start of triangle preceded by triangle "A213998/A213999":
. 0:                      1                                 1
. 1:                   1   1/2                             1  0
. 2:               1    3/2   1/3                         1  1  0
. 3:            1   5/2    11/6   1/4                    1 2  1  0
. 4:        1   7/2   13/3    25/12   1/5               1 3  4  2 0
. 5:     1   9/2   47/6   77/12   137/60   1/6         1 4  7  6 2 0
. 6:  1  11/2   37/3   57/4   87/10   49/20   1/7,    1 5 12 14 8 2 0.

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened

a214075 n k = a214075_tabl !! n !! k
a214075_row n = a214075_tabl !! n
a214075_tabl = zipWith (zipWith div) a213998_tabl a213999_tabl

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

A349257 Largest integer that can be expressed as Sum_{k=1..n} k/p(k), where p is a permutation of [n].

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 11, 15, 18, 21, 22, 27, 28, 32, 36, 40, 41, 46, 47

Offset: 0

Seiichi Manyama, Nov 12 2021

Cf. A022819, A073090, A349277.

def A(n)
  max = 0
  (1..n).to_a.permutation{|i|
    m = (1..n).inject(0){|s, j| s + j / i[j - 1].to_r}
    if m.denominator == 1
      max = m if max < m
    end
  }
  max.to_i
end
def A349257(n)
  (0..n).map{|i| A(i)}
end
p A349257(8)

a(n) = 1 + a(n-1) if n is prime. - Alois P. Heinz, Nov 12 2021

a(12)-a(19) from Alois P. Heinz, Nov 12 2021

A225154 Floor(Sum_{i=1..n} (Sum_{j=1..i} sqrt(1/j))).

Original entry on oeis.org

1, 2, 4, 7, 11, 14, 18, 23, 27, 32, 38, 43, 49, 55, 62, 68, 75, 82, 90, 97, 105, 113, 121, 130, 138, 147, 156, 166, 175, 185, 194, 204, 214, 225, 235, 246, 257, 267, 279, 290, 301, 313, 325, 336, 349, 361, 373, 385, 398

Offset: 1

Balarka Sen, Apr 30 2013

The fact that a(n)/n diverges (it is greater than sqrt(n)) implies sum_{k>=1} 1/sqrt(k) is not Cesaro summable.

Balarka Sen, Table of n, a(n) for n = 1..500

Cf. A022819, A025224.

for(n=1,100,print1(floor(sum(i=1,n,sum(j=1,i,1/sqrt(j))))","))

a(n)=sum(j=1,n,(n+1-j)/sqrt(j))\1 \\ Charles R Greathouse IV, May 02 2013

a(n) ~ 2*Sum_{k=1..n} sqrt(k) ~ (4/3) n^(3/2).

cp's OEIS Frontend

A081527 Erroneous version of A022819.

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A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.

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A214075 Triangle read by rows: T(n,k) = floor(A213998(n,k) / A213999(n,k)), 0<=k<=n.

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Haskell

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

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A349257 Largest integer that can be expressed as Sum_{k=1..n} k/p(k), where p is a permutation of [n].

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A225154 Floor(Sum_{i=1..n} (Sum_{j=1..i} sqrt(1/j))).

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PARI

PARI

Formula