A027612 -id:A027612 - cp's OEIS frontend

A213998 Numerators of the triangle of fractions read by rows: pf(n,0) = 1, pf(n,n) = 1/(n+1) and pf(n+1,k) = pf(n,k) + pf(n,k-1) with 0 < k < n.

1, 1, 1, 1, 3, 1, 1, 5, 11, 1, 1, 7, 13, 25, 1, 1, 9, 47, 77, 137, 1, 1, 11, 37, 57, 87, 49, 1, 1, 13, 107, 319, 459, 223, 363, 1, 1, 15, 73, 533, 743, 341, 481, 761, 1, 1, 17, 191, 275, 1879, 2509, 3349, 4609, 7129, 1, 1, 19, 121, 1207, 1627, 2131, 2761, 3601, 4861, 7381, 1

Offset: 0

Reinhard Zumkeller, Jul 03 2012

T(n,0) = 1;
T(n,1) = A005408(n-1) for n > 0;
T(n,2) = A188386(n-2) for n > 2;
T(n,n-3) = A124837(n-2) for n > 2;
T(n,n-2) = A027612(n-1) for n > 1;
T(n,n-1) = A001008(n) for n > 0;
T(n,n) = 1;
A214075(n,k) = floor(T(n,k) / A213999(n,k)).

			Start of triangle pf with corresponding triangles of numerators and denominators:
. 0:                            1
. 1:                         1    1/2
. 2:                     1     3/2    1/3
. 3:                  1    5/2    11/6    1/4
. 4:              1   7/2    13/3    25/12    1/5
. 5:           1    9/2   47/6    77/12   137/60   1/6
. 6:        1  11/2   37/3    57/4    87/10    49/20    1/7
. 7:     1  13/2  107/6  319/12  459/20   223/20  363/140   1/8
. 8:  1  15/2  73/3  533/12  743/15  341/10   481/35   761/280  1/9,
.
. 0:   numerators     1                          1    denominators
. 1:                1  1                        1  2       A213999
. 2:              1   3  1                     1 2  3
. 3:            1   5  11 1                   1 2 6  4
. 4:          1  7  13  25  1                1 2 3  12 5
. 5:        1  9  47  77 137  1             1 2 6 12  60 6
. 6:      1 11  37 57  87  49  1           1 2 3 4 10  20  7
. 7:    1 13 107 319 459 223 363 1        1 2 6 12 20 20 140 8
. 8:  1 15 73 533 743 341 481 761 1,     1 2 3 12 15 10 35 280 9.

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A005408, A188386 (columns).
Cf. A001008, A027612, A124837 (diagonals).
Cf. A213999 (denominators).

import Data.Ratio ((%), numerator, denominator, Ratio)
a213998 n k = a213998_tabl !! n !! k
a213998_row n = a213998_tabl !! n
a213998_tabl = map (map numerator) $ iterate pf [1] where
   pf row = zipWith (+) ([0] ++ row) (row ++ [-1 % (x * (x + 1))])
            where x = denominator $ last row

T[, 0] = 1; T[n, n_] := 1/(n + 1);
T[n_, k_] := T[n, k] = T[n - 1, k] + T[n - 1, k - 1];
Table[T[n, k] // Numerator, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2021 *)

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

A115071 Numerator of 1^n/n + 2^n/(n-1) + 3^n/(n-2) +...+ (n-1)^n/2 + n^n/1.

Original entry on oeis.org

1, 9, 94, 3625, 18631, 1120581, 34793764, 5692787001, 29669041771, 30708223774261, 134127439064434, 302304605103335861, 2387352152511746837, 109134149200789179825, 24217460586461892638584

Offset: 1

Alexander Adamchuk, Jun 17 2006

a(p-1) is divisible by p^3 for prime p>3. a(p^2-1) is divisible by p^6 for prime p>3. a(p^3-1) is divisible by p^9 for prime p>3.

			a(1) = numerator(1/1) = 1.
a(2) = numerator(1/2 + 4/1) = 9.
a(3) = numerator(1/3 + 8/2 + 27/1) = 94.
a(4) = numerator(1/4 + 16/3 + 81/2 + 256/1) = 3625.

Cf. A001008, A027612, A120487 (denominator).

Table[Numerator[Sum[i^n/(n+1-i),{i,1,n}]],{n,1,20}]

a(n) = numerator(Sum_{i=1..n} i^n/(n+1-i)).

A124838 Denominators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Original entry on oeis.org

1, 2, 6, 4, 20, 10, 70, 56, 504, 420, 4620, 3960, 3432, 6006, 90090, 80080, 1361360, 408408, 369512, 67184, 470288, 1293292, 29745716, 27457584, 228813200, 212469400, 5736673800, 5354228880, 155272637520, 291136195350, 273491577450

Offset: 1

Jonathan Vos Post, Nov 10 2006

Numerators are A124837. All fractions reduced. Thanks to Jonathan Sondow for verifying these calculations. He suggests that the equivalent definition in terms of first order harmonic numbers may be computationally simpler. We are happy with the description of A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n, but baffled by the description of A027611.

			a(1) = 1 = denominator of 1/1.
a(2) = 2 = denominator of 1/1 + 5/2 = 7/2.
a(3) = 6 = denominator of 7/2 + 13/3 = 47/6.
a(4) = 4 = denominator of 47/6 + 77/12 = 57/4.
a(5) = 20 = denominator of 57/4 + 87/10 = 549/20.
a(6) = 10 = denominator of 549/20 + 223/20 = 341/10
a(7) = 70 = denominator of 341/10 + 481/35 = 3349/70.
a(8) = 1260 = denominator of 3349/70 + 4609/280 = 88327/1260.
a(9) = 45 = denominator of 88327/1260 + 4861/252 = 3844/45.
a(10) = 504 = denominator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
a(10) = 504 = denominator of 1/1 + 5/2 + 13/3 + 77/12 + 87/10 + 223/20 + 481/35 + 4609/252 + 4861/252 + 55991/2520 = 54251/504.

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Harmonic Number. See equation for third-order harmonic numbers.

Cf. A027611, A027612, A124837.

a124838 n = a213999 (n + 2) (n - 1) -- Reinhard Zumkeller, Jul 03 2012

Table[Denominator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,40}] (* Alexander Adamchuk, Nov 11 2006 *)

A124837(n)/A124838(n) = Sum_{i=1..n} A027612(n)/A027611(n+1).
a(n) = denominator(Sum_{m=1..n} Sum_{L=1..m} Sum_{k=1..L} 1/k).
a(n) = denominator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k). - Alexander Adamchuk, Nov 11 2006
a(n) = A213999(n+2,n-1). - Reinhard Zumkeller, Jul 03 2012

Corrected and extended by Alexander Adamchuk, Nov 11 2006

A363000 a(n) = numerator(R(n, n, 1)), where R are the rational polynomials R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Original entry on oeis.org

1, 5, 19, 188, 1249, 125744, 283517, 303923456, 138604561, 599865008128, 118159023973, 7078040993755136, 155792758736921, 146303841678548271104, 294014633772018349, 64670474732430319157248, 752324747622089633569, 3224753626003393505960919040, 2507759850059601711479669

Offset: 0

Peter Luschny, May 12 2023

R(n, n, 0) are the (0-based) harmonic numbers, R(n, n, -1) are the Bernoulli numbers, and R(n, n, 1) is this sequence in its rational form.

			a(n) are the numerators of the terms on the main diagonal of the triangle:
[0] 1;
[1] 1,  5/2;
[2] 1,  7/2,  19/2;
[3] 1, 11/2, 121/6,  188/3;
[4] 1, 19/2,  95/2,  369/2,   1249/2;
[5] 1, 35/2, 721/6, 1748/3, 35164/15, 125744/15;
[6] 1, 67/2, 639/2, 3877/2,  18533/2,   76317/2, 283517/2;

Cf. A363001 (denominators), A362999 (odd-indexed denominators), A362998.

Cf. A027611, A027612, A062709.

# For better context we put A362998, A362999, A363000, and A363001 together here.
R := (n, k, x) -> add(add(x^j*binomial(u, j)*(j+1)^n, j=0..u)/(u + 1), u=0..k):
### x = 1 -> this sequence
 for n from 0 to 7 do [n], seq(R(n, k, 1), k = 0..n) od;
 seq(R(n, n, 1), n = 0..9);
 A363000 := n -> numer(R(n, n, 1)): seq(A363000(n), n = 0..10);
 A363001 := n -> denom(R(n, n, 1)): seq(A363001(n), n = 0..20);
 A362999 := n -> denom(R(2*n+1, 2*n+1, 1)): seq(A362999(n), n = 0..11);
 A362998 := n -> add(R(2*n, k, 1), k = 0..2*n): seq(A362998(n), n = 0..9);
### x = -1 -> Bernoulli(n, 1)
# for n from 0 to 9 do [n], seq(R(n, k,-1), k = 0..n) od;
# seq(R(n, n, -1), n = 0..12); seq(bernoulli(n, 1), n = 0..12);
### x = 0 -> Harmonic numbers
# for n from 0 to 9 do [n], seq(R(n, k, 0), k = 0..n) od;
# seq(R(n, n, 0), n = 0..9); seq(harmonic(n+1), n = 0..9);

Sum_{k=0..n} R(n, k, 0) = Sum_{j=0..n} (n-j+1)/(j+1) = (n+2)*Harmonic(n+1)-n-1.
Sum_{k=0..n} R(n, k,-1) = (n + 2 - 0^n) * Bernoulli(n, 1).
Sum_{k=0..2*n} R(2*n, k, 1) = A362998(n).
2*R(n, 1, 1) = A062709(n).

A124880 Primes in A124837.

Original entry on oeis.org

7, 47, 42131, 23763863, 192066102203, 5733412167187, 34745876421709, 185813891783454008069, 171312804637561107990389, 29207630124216024960052176833, 6300447575454970515437116064749

Offset: 1

Alexander Adamchuk, Nov 11 2006

nonn

			A124837(n) begins {1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, ...}.
Thus a(1) = 7, a(2) = 47, a(3) = 42131.

Eric Weisstein's World of Mathematics, Harmonic Number.

A124837 are the numerators of third-order harmonic numbers H(n, (3)).
Corresponding numbers n such that A124837(n) is prime are listed in A124881.
Cf. A001008, A002805, A067657, A056903, A027612, A124878, A124879, A124837, A124881.

s=3/2;Do[s=s+1/n;f=Numerator[n*(n-1)/2*(s-3/2)]; If[PrimeQ[f],Print[{n-2,f}]],{n,3,125}]

Crossrefs edited by Michel Marcus, Jul 14 2018

A124881 Numbers k such that A124837(k) is prime.

Original entry on oeis.org

2, 3, 9, 15, 25, 27, 33, 45, 55, 67, 70, 93, 94, 97, 112, 113, 125, 137, 189, 193, 212, 232, 262, 273, 281, 381, 453, 528, 670, 677, 742, 743, 827, 996, 1257, 1349, 1402, 1645, 1683, 2110, 2217, 2408, 2480, 2623, 3208, 3517, 3637, 3665, 4571, 4730

Offset: 1

Alexander Adamchuk, Nov 11 2006

			A124837(n) begins {1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, ...}.
Thus a(1) = 2, a(2) = 3, a(3) = 9.

Eric Weisstein's World of Mathematics, Harmonic Number.

A124837 are the numerators of third-order harmonic numbers H(n, (3)).
Corresponding primes in A124837 are listed in A124880.
Cf. A001008, A002805, A067657, A056903, A027612, A124878, A124879, A124837, A124880.

s=3/2;Do[s=s+1/n;f=Numerator[n*(n-1)/2*(s-3/2)]; If[PrimeQ[f],Print[{n-2,f}]],{n,3,1000}]

More terms from Stefan Steinerberger, May 09 2007

Crossrefs edited by Michel Marcus, Jul 14 2018

A120286 Numerator of 1/n^2 + 2/(n-1)^2 + 3/(n-2)^2 +...+ (n-1)/2^2 + n.

Original entry on oeis.org

1, 9, 65, 725, 3899, 28763, 419017, 864669, 7981633, 3586319, 200763407, 2649665993, 34899471137, 176508049513, 356606957297, 12234391348253, 209672027529221, 4012917216669239, 15350275129353301, 15443118015171841

Offset: 1

Alexander Adamchuk, Jul 07 2006

p^2 divides a(p-1) for prime p>2. p divides a(p-2) for prime p>3.

Cf. A027612, A001008, A007406, A007407, A370774 (denom.).

Numerator[Table[Sum[Sum[1/i^2,{i,1,k}],{k,1,n}],{n,1,30}]]
Table[-EulerGamma + HarmonicNumber[1 + n, 2] + n*HarmonicNumber[1 + n, 2] - PolyGamma[0, 2 + n], {n, 1, 20}] // Numerator (* Vaclav Kotesovec, May 02 2024 *)

from fractions import Fraction
def A120286(n): return sum(Fraction(n-i+1,i**2) for i in range(1,n+1)).numerator # Chai Wah Wu, May 01 2024

a(n) = numerator[Sum[Sum[1/i^2,{i,1,k}],{k,1,n}]].

A160048 Numerator of the Harary number for the path graph P_n.

Original entry on oeis.org

0, 2, 5, 26, 77, 87, 223, 962, 4609, 4861, 55991, 58301, 785633, 811373, 835397, 3431678, 29889983, 30570663, 197698279, 201578155, 41054655, 13920029, 325333835, 990874363, 25128807667, 25472027467, 232222818803, 235091155703

Offset: 1

Eric W. Weisstein, Apr 30 2009

			0, 2, 5, 26/3, 77/6, 87/5, 223/10, 962/35, 4609/140, 4861/126, ...

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Harary Index

Cf. A160049, A027612.

harmonic(n)=sum(k=1,n,1/k);
a(n)=numerator(2*n*(harmonic(n)-1)); \\ Andrew Howroyd, Oct 31 2017

A354859 a(n) is the numerator of 1/prime(n) + 2/prime(n-1) + 3/prime(n-2) + ... + (n-1)/3 + n/2.

Original entry on oeis.org

1, 4, 71, 124, 11111, 92402, 257189, 43708274, 2344987003, 1855369084, 2729707472269, 11281836318542, 1705853427969059, 120757830191824486, 1124815045783478971, 118422287191742563724, 1291008724583331399881, 113743044027018860349034, 70575236921156825443680027, 8002471039307070610187173702

Offset: 1

Ilya Gutkovskiy, Jun 09 2022

Numerator of a second order prime harmonic number.

			1/2, 4/3, 71/30, 124/35, 11111/2310, 92402/15015, 257189/34034, ...

Cf. A024451, A027612, A120272, A354860 (denominators).

Table[Sum[(n - k + 1)/Prime[k], {k, 1, n}], {n, 1, 20}] // Numerator

from fractions import Fraction
from sympy import prime, primerange
def a(n): return sum(Fraction(n-i, p) for i, p in enumerate(primerange(1, prime(n)+1))).numerator
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 09 2022

a(n) is the numerator of Sum_{j=1..n} Sum_{i=1..j} 1/prime(i).

cp's OEIS Frontend

A213998 Numerators of the triangle of fractions read by rows: pf(n,0) = 1, pf(n,n) = 1/(n+1) and pf(n+1,k) = pf(n,k) + pf(n,k-1) with 0 < k < n.

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

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A115071 Numerator of 1^n/n + 2^n/(n-1) + 3^n/(n-2) +...+ (n-1)^n/2 + n^n/1.

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A124838 Denominators of third-order harmonic numbers (defined by Conway and Guy, 1996).

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A363000 a(n) = numerator(R(n, n, 1)), where R are the rational polynomials R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

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A124880 Primes in A124837.

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A124881 Numbers k such that A124837(k) is prime.

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A120286 Numerator of 1/n^2 + 2/(n-1)^2 + 3/(n-2)^2 +...+ (n-1)/2^2 + n.