A098464 -id:A098464

A110566

a(n) = lcm{1,2,...,n}/denominator of harmonic number H(n).

+10
24

1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 15, 45, 45, 45, 15, 3, 3, 1, 1, 1, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 77, 77, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 9, 9, 9, 27, 27, 27, 9, 9, 9, 3, 3, 3, 3, 3, 33, 33, 33, 33, 11, 11, 11, 11, 11, 11, 11, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

a(n) is always odd.

Unsorted union: 1, 3, 15, 45, 11, 77, 7, 9, 27, 33, 25, 5, 55, 275, 13, 39, 17, 49, 931, 19, 319, 75, ..., . See A112810.

It is conjectured that every odd number occurs in this sequence (see A112822 for the first occurrence of each of them). - Jianing Song, Nov 28 2022

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A003418(n)/A002805(n) = A025529(n)/A001008(n).

From Franz Vrabec, Sep 21 2005: (Start)

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

a(n) = gcd(A003418(n), A025529(n)). (End)

EXAMPLE

a(6) = 60/20 = 3 because lcm{1,2,3,4,5,6}=60 and H(6)=49/20.

MAPLE

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:

L:= proc(n) L(n):= ilcm(n, `if`(n=1, 1, L(n-1))) end:

a:= n-> L(n)/denom(H(n)):

seq(a(n), n=1..100); # Alois P. Heinz, Aug 30 2012

MATHEMATICA

f[n_] := LCM @@ Range[n]/Denominator[HarmonicNumber[n]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Sep 15 2005 *)

PROG

(PARI) a(n) = lcm(vector(n, k, k))/denominator(sum(k=1, n, 1/k)); \\ Michel Marcus, Mar 07 2018

(Python)

from sympy import lcm, harmonic

def A110566(n): return lcm([k for k in range(1, n+1)])//harmonic(n).q # Chai Wah Wu, Mar 06 2021

CROSSREFS

Cf. A001008, A002805, A003418, A025529, A098464, A112810, A112822, A358557.

KEYWORD

nonn

AUTHOR

Franz Vrabec, Sep 12 2005

EXTENSIONS

More terms from Robert G. Wilson v, Sep 15 2005

STATUS

approved

A112822

Least number k such that lcm{1,2,...,k}/denominator of harmonic number H(k) = 2n-1.

+10
14

1, 6, 105, 44, 63, 33, 156, 20, 272, 343, 38272753, 11881, 100, 66, 822, 28861, 77

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

First occurrence of 2n-1 in A110566.

Sequence continues: a(18)=?, 1332, 162, 2758521, 24649, 21, a(24)=?, 294, a(26)=?, 1166, 110, 126059, 201957, 3660, 37553041, 344929, 296341, a(35)=?, 25155299, a(37)=?, 500, 42

LINKS

Table of n, a(n) for n=1..17.

MATHEMATICA

a = h = 1; t = Table[0, {100}]; Do[a = LCM[a, n]; h = h + 1/n; b = a/Denominator[h]; If[b < 101 && t[[(b + 1)/2]] == 0, t[[(b + 1)/2]] = n], {n, 500000}]; t

PROG

(Python)

from fractions import Fraction

from sympy import lcm

def A112822(n):

k, l, h = 1, 1, Fraction(1, 1)

while l != h.denominator*(2*n-1):

k += 1

l = lcm(l, k)

h += Fraction(1, k)

return k # Chai Wah Wu, Mar 06 2021

CROSSREFS

Cf. A110566, A098464, A112813, A112814, A112815, A112816, A112817, A112818, A112819, A112820, A112821.

KEYWORD

nonn,more

AUTHOR

Robert G. Wilson v, Sep 15 2005

EXTENSIONS

a(11), a(32) from Max Alekseyev, Nov 29 2013

a(33)-a(34) from Chai Wah Wu, Mar 06 2021

a(36), a(38), a(39) from Chai Wah Wu, Mar 12 2021

STATUS

approved

A112813

Numbers k such that lcm(1,2,3,...,k)/3 equals the denominator of the k-th harmonic number H(k).

+10
13

6, 7, 8, 18, 19, 25, 26, 54, 55, 56, 57, 58, 59, 60, 61, 62, 72, 73, 74, 75, 76, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 3 occurs in A110566.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

MATHEMATICA

f[n_] := LCM @@ Range[n]/Denominator[ HarmonicNumber[n]]; Select[ Range[231], f[ # ] == 3 &]

PROG

(PARI) isok(n) = lcm(vector(n, i, i)) == 3*denominator(sum(i=1, n, 1/i)); \\ Michel Marcus, Mar 07 2018

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112814, A112815, A112816, A112817, A112818, A112819, A112820, A112821, A112822.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

STATUS

approved

A112814

Numbers k such that lcm(1,2,3,...,k)/5 equals the denominator of the k-th harmonic number H(k).

+10
12

105, 106, 107, 108, 109, 2625, 2626, 2627, 2628, 2629, 2630, 2631, 2632, 2633, 2634, 2635, 2636, 2637, 2638, 2639, 2640, 2641, 2642, 2643, 2644, 2645, 2646, 2647, 2648, 2649, 2650, 2651, 2652, 2653, 2654, 2655, 2656, 2657, 2658, 2659, 2660, 2661, 2662

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 5 occurs in A110566.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

MATHEMATICA

f[n_] := LCM @@ Range[n]/Denominator[ HarmonicNumber[n]]; Select[ Range[2662], f[ # ] == 5 &]

PROG

(PARI) isok(n) = lcm(vector(n, i, i)) == 5*denominator(sum(i=1, n, 1/i)); \\ Michel Marcus, Mar 07 2018

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112815, A112816, A112817, A112818, A112819, A112820, A112821, A112822.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

STATUS

approved

A112815

Numbers k such that lcm(1,2,3,...,k)/7 equals the denominator of the k-th harmonic number H(k).

+10
12

44, 45, 46, 47, 48, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 2209, 2210, 2211, 2212, 2213, 2214, 2215, 2216, 2217, 2218, 2219

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 7 occurs in A110566.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

MATHEMATICA

f[n_] := LCM @@ Range[n]/Denominator[ HarmonicNumber[n]]; Select[ Range[2219], f[ # ] == 7 &]

PROG

(PARI) isok(n) = lcm(vector(n, i, i)) == 7*denominator(sum(i=1, n, 1/i)); \\ Michel Marcus, Mar 07 2018

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112816, A112817, A112818, A112819, A112820, A112821, A112822.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

STATUS

approved

A112816

Numbers k such that lcm(1,2,3,...,k)/9 equals the denominator of the k-th harmonic number H(k).

+10
12

63, 64, 65, 69, 70, 71, 189, 190, 191, 192, 193, 194, 195, 196, 197, 207, 208, 209, 210, 211, 212, 213, 214, 215, 1701, 1702, 1703, 1704, 1705, 1706, 1707, 1708, 1709, 1710, 1711, 1712, 1713, 1714, 1715, 1716, 1717, 1718, 1719, 1720, 1721, 1722, 1723, 1724

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 9 occurs in A110566.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1483 from Jinyuan Wang)

MATHEMATICA

f[n_] := LCM @@ Range[n]/Denominator[ HarmonicNumber[n]]; Select[ Range[1724], f[ # ] == 9 &]

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112817, A112818, A112819, A112820, A112821, A112822.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

EXTENSIONS

Definition corrected by Jinyuan Wang, May 03 2020

STATUS

approved

A112817

Numbers k such that lcm(1,2,3,...,k)/11 equals the denominator of the k-th harmonic number H(k).

+10
12

33, 34, 35, 36, 37, 38, 39, 40, 41, 81, 82, 83, 84, 85, 86, 87, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 11 occurs in A110566.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jinyuan Wang)

MATHEMATICA

f[n_] := LCM @@ Range[n]/Denominator[ HarmonicNumber[n]]; Select[ Range[430], f[ # ] == 11 &]

Select[Range[450], 1/11*LCM@@Range[#]==Denominator[HarmonicNumber[#]]&] (* Harvey P. Dale, Jan 06 2019 *)

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112816, A112818, A112819, A112820, A112821, A112822.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

EXTENSIONS

Name (definition) corrected by Harvey P. Dale, Jan 06 2019

STATUS

approved

A112818

Numbers k such that lcm(1,2,3,...,k)/13 equals the denominator of the k-th harmonic number H(k).

+10
12

156, 157, 158, 159, 160, 161, 27380, 27381, 27382, 27383, 27384, 27385, 27386, 27387, 27388, 27389, 27390, 27391, 27392, 27393, 27394, 27395, 27396, 27397, 27398, 27399, 27400, 27401, 27402, 27403, 27404, 27405, 27406, 27407, 27408

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 13 occurs in A110566.

LINKS

Jinyuan Wang, Table of n, a(n) for n = 1..851

MATHEMATICA

a = h = 1; t = {}; Do[a = LCM[a, n]; h = h + 1/n; b = a/Denominator[h]; If[b == 13, AppendTo[t, n]], {n, 27408}]; t

With[{tk=Table[{LCM@@Range[k]/13, Denominator[HarmonicNumber[k]]}, {k, 28000}]}, Position[ tk, _?(#[[1]]==#[[2]]&), 1, Heads->False]]//Flatten (* Harvey P. Dale, Apr 02 2022 *)

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112816, A112817, A112819, A112820, A112821, A112822.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

EXTENSIONS

Definition corrected by Jinyuan Wang, May 03 2020

STATUS

approved

A112819

Numbers k such that lcm(1,2,3,...,k)/15 equals the denominator of the k-th harmonic number H(k).

+10
12

20, 24, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 41889597, 41889598, 41889599, 41889600, 41889601, 41889602, 41889603, 41889604, 41889605, 41889606, 41889607

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 15 occurs in A110566.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

MATHEMATICA

a = h = 1; t = {}; Do[a = LCM[a, n]; h = h + 1/n; If[a/Denominator[h] == 15, AppendTo[t, n]], {n, 10^6}]; t

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112816, A112817, A112818, A112820, A112821, A112822.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

EXTENSIONS

Definition corrected by Jinyuan Wang, May 03 2020

More terms from Chai Wah Wu, Mar 18 2021

STATUS

approved

A112820

Numbers k such that lcm(1,2,3,...,k)/17 equals the denominator of the k-th harmonic number H(k).

+10
12

272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 79507, 79508, 79509, 79510, 79511, 79512, 79513, 79514, 79515, 79516, 79517, 79518, 79519, 79520, 79521, 79522, 79523, 79524, 79525, 79526, 79527, 79528

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When 17 occurs in A110566.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..3106 from Jinyuan Wang)

MATHEMATICA

a = h = 1; t = {}; Do[a = LCM[a, n]; h = h + 1/n; If[a/Denominator[h] == 17, AppendTo[t, n]], {n, 79528}]; t

CROSSREFS

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112816, A112817, A112818, A112819, A112821, A112822.