A124837 -id:A124837 - cp's OEIS frontend

A124880 Primes in A124837.

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

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.

Original entry on oeis.org

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 *)

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

A124878 Primes in A027612.

Original entry on oeis.org

5, 13, 223, 4861, 197698279, 25472027467, 6975593267347, 218572480850557, 1592457339642613, 2955634782407818711841368777079578319, 2950127241932882597818337002939124083061, 232242878286351670588710938679161483012314573381293769

Offset: 1

Alexander Adamchuk, Nov 11 2006

nonn

			A027612(n) begins {1, 5, 13, 77, 87, 223, 481, 4609, 4861, ...}.
Thus a(1) = 5, a(2) = 13, a(3) = 223, a(4) = 4861.

Eric Weisstein's World of Mathematics, Harmonic Number.

A027612(n) are the numerators of second order harmonic numbers H(n, (2)).
Corresponding numbers n such that A027612(n) is prime are listed in A124879.
Cf. A001008, A002805, A067657, A056903, A027612, A124879, A124837, A124880, A124881.

s=1;Do[s=s+1/(n+1);f=Numerator[(n+1)*(s-1)]; If[PrimeQ[f],Print[{n,f}]],{n,1,1942}]

lista(nn) = {for (n=1, nn, if (isprime(p=numerator(sum(k=1, n, k/(n-k+1)))), print1(p, ", ")););} \\ Michel Marcus, Jul 14 2018

a(n) = A027612(A124879(n)).

a(12) from, and crossrefs edited by Michel Marcus, Jul 14 2018

A124879 Numbers k such that A027612(k) is prime.

Original entry on oeis.org

2, 3, 6, 9, 18, 25, 29, 30, 39, 84, 91, 125, 130, 184, 195, 199, 203, 241, 245, 273, 281, 378, 552, 571, 653, 776, 901, 1099, 1215, 1224, 1235, 1315, 1412, 1657, 1942, 2076, 2085, 2743, 2745, 2855, 2859, 3517, 3717, 4183, 4188, 4362, 4547, 4728, 4783

Offset: 1

Alexander Adamchuk, Nov 11 2006

nonn

			A027612 begins {1, 5, 13, 77, 87, 223, 481, 4609, 4861, ...}.
Thus a(1) = 2, a(2) = 3, a(3) = 6, a(4) = 9.

Eric Weisstein's World of Mathematics, Harmonic Number.

A027612(n) are the numerators of second order harmonic numbers H(n, (2)).
Corresponding primes in A027612 are listed in A124878.
Cf. A001008, A002805, A067657, A056903, A027612, A124878, A124837, A124880, A124881.

s=1;Do[s=s+1/(n+1);f=Numerator[(n+1)*(s-1)]; If[PrimeQ[f],Print[{n,f}]],{n,1,1942}]

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

More terms from Stefan Steinerberger, May 29 2007

Crossrefs edited by Michel Marcus, Jul 14 2018

A076174 Numerator of Sum_{i+j+k=n, i,j,k>=1} (i*j)/k.

Original entry on oeis.org

0, 0, 1, 9, 37, 319, 743, 2509, 2761, 32891, 35201, 485333, 511073, 535097, 1115239, 19679783, 6786821, 133033679, 136913555, 140608675, 144135835, 678544345, 693417203, 17692378667, 18035598467, 165294957803, 168163294703

Offset: 1

Benoit Cloitre, Nov 01 2002

a(n) is odd.

a(n+2) = Numerators of 4th-order harmonic numbers (defined by Conway and Guy, 1996). - Alexander Adamchuk, Jun 14 2008

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

Alexander Adamchuk, Jun 14 2008, Table of n, a(n) for n = 1..52

Cf. A076175.

Cf. A124837 = Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Table[ Numerator[Sum[ Sum[ Sum[ Sum[ 1/k, {k,1,l} ], {l,1,m} ], {m,1,n} ], {n,1,s-2} ] ], {s,1,52} ] Table[ Numerator[ (n-1)n(n+1)/6 * Sum[ 1/k, {k,4,n+1} ] ], {n,1,50}] (* Alexander Adamchuk, Jun 14 2008 *)

a(n)=numerator(sum(i=1,n,sum(j=1,n,sum(k=1,n,if(n-i-j-k,0,1)*i*j/k))))

a(n) = Numerator[Sum[ Sum[ Sum[ Sum[ 1/k, {k,1,l} ], {l,1,m} ], {m,1,n} ], {n,1,s-2} ] ]. a(n) = Numerator[ (n-1)n(n+1)/6 * Sum[ 1/k, {k,4,n+1} ] ]. - Alexander Adamchuk, Jun 14 2008

a(n) = Numerator(sum(1/(k+3), k=1..n-2)), n>1. - Gary Detlefs, Sep 14 2011

A354894 a(n) is the numerator of the n-th hyperharmonic number of order n.

Original entry on oeis.org

1, 5, 47, 319, 1879, 20417, 263111, 261395, 8842385, 33464927, 166770367, 3825136961, 19081066231, 57128792093, 236266661971, 7313175618421, 14606816124167, 102126365345729, 3774664307989373, 3771059091081773, 154479849447926113, 6637417807457499259, 6632660439700528339

Offset: 1

Ilya Gutkovskiy, Jun 10 2022

			1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...

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

Eric Weisstein's World of Mathematics, Harmonic Number
Wikipedia, Hyperharmonic number

Cf. A001008, A001700, A027612, A058806, A076174, A124837, A354895 (denominators).

Differs from A049281.

Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 23}] // Numerator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 23}] // Numerator

H(n) = sum(i=1, n, 1/i);
a(n) = numerator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ Michel Marcus, Jun 10 2022

from math import comb
from sympy import harmonic
def A354894(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).p # Chai Wah Wu, Jun 18 2022

a(n) is the numerator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
a(n) is the numerator of binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1)), where H(n) is the n-th harmonic number.
a(n) / A354895(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).

A316297 a(n) = n! times the denominator of the n-th harmonic number H(n).

Original entry on oeis.org

1, 4, 36, 288, 7200, 14400, 705600, 11289600, 914457600, 9144576000, 1106493696000, 13277924352000, 2243969215488000, 31415569016832000, 471233535252480000, 15079473128079360000, 4357967734014935040000, 26147806404089610240000, 9439358111876349296640000

Offset: 1

Matthew Campbell, Jun 29 2018

nonn

			a(4) = 4! * A002805(4) = 24 * 12 = 288.

Cf. A000142, A001008, A002805, A027611, A027612, A124837, A124838.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> denom(H(n))*n!:
seq(a(n), n=1..20);  # Alois P. Heinz, Jul 21 2018

a[n_] := n! Denominator@HarmonicNumber@n; Array[a, 18] (* Robert G. Wilson v, Jun 30 2018 *)

a(n) = n! * denominator(sum(k=1, n, 1/k)); \\ Michel Marcus, Aug 12 2018

a(n) = A000142(n) * A002805(n).

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

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A124878 Primes in A027612.

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A124879 Numbers k such that A027612(k) is prime.

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A076174 Numerator of Sum_{i+j+k=n, i,j,k>=1} (i*j)/k.

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