A110566 -id:A110566 - cp's OEIS frontend

A112814 Numbers k such that lcm(1,2,3,...,k)/5 equals the denominator of the k-th harmonic number H(k).

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 5 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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

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

A112815 Numbers k such that lcm(1,2,3,...,k)/7 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 7 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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

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

A112816 Numbers k such that lcm(1,2,3,...,k)/9 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 9 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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

Definition corrected by Jinyuan Wang, May 03 2020

A112817 Numbers k such that lcm(1,2,3,...,k)/11 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 11 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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 *)

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

A112818 Numbers k such that lcm(1,2,3,...,k)/13 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 13 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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 *)

Definition corrected by Jinyuan Wang, May 03 2020

A112819 Numbers k such that lcm(1,2,3,...,k)/15 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 15 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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

Definition corrected by Jinyuan Wang, May 03 2020

More terms from Chai Wah Wu, Mar 18 2021

A112820 Numbers k such that lcm(1,2,3,...,k)/17 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 17 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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

Definition corrected by Jinyuan Wang, May 03 2020

A112821 Numbers n such that lcm(1,2,3,...,n)/19 equals the denominator of the n-th harmonic number H(n).

Original entry on oeis.org

343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360

Offset: 1

Robert G. Wilson v, Sep 17 2005

Positions where 19 occurs in A110566.

Tanya Khovanova, Non Recursions

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112816, A112817, A112818, A112819, A112820, A112822.

a = h = 1; t = {}; Do[a = LCM[a, n]; h = h + 1/n; If[a/Denominator[h] == 19, AppendTo[t, n]], {n, 10^6}]; t
Select[Range[500],Denominator[HarmonicNumber[#]]==LCM@@Range[#]/19&] (* Harvey P. Dale, Jan 29 2012 *)

Definition corrected by Max Alekseyev, Mar 03 2007

A342350 Numbers k such that lcm(1,2,3,...,k)/21 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

38272753, 38272754, 38272755, 38272756, 38272757, 38272758, 38272759, 38272760, 38272761, 38272762, 38272763, 38272764, 38272765, 38272766, 38272767, 38272768, 38272769, 38272770, 38272771, 38272772, 38272773, 38272774, 38272775, 38272776, 38272777, 38272778

Offset: 1

Chai Wah Wu, Mar 17 2021

nonn

Positions where 21 occurs in A110566.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Tanya Khovanova, Non Recursions

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112816, A112817, A112818, A112819, A112820, A112821, A112822.

A358557 Numbers k for which denominator(H(k)) < LCM(1..k), where harmonic numbers H(k) = Sum_{i=1..k} 1/i = r(k)/q(k).

Original entry on oeis.org

6, 7, 8, 18, 19, 20, 21, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 100, 101, 102

Offset: 1

Yifan Xie, Nov 22 2022

nonn

LCM(1..k) is a common denominator for the harmonic numbers, and the present terms k are where the sum reduces to a smaller denominator (A002805).
We can find a prime p and a pair of positive integers t < p and o for every positive integer k that p^o*t <= k < p^o*(t+1). For positive integers i that are not divisible by p^o, a multiple of p will be added to the numerator of the reciprocal sum; for i's that are divisible by p^o, the number that will be added to the numerator of the reciprocal sum is divisible by r(t). So k is in the sequence if and only if p^o*t <= k < p^o*(t+1) where p is a prime and p divides r(t).
The sequence is the answer to Problem 23 of the 2022 AMC12A.

Yifan Xie, Table of n, a(n) for n = 1..10000
Art of Problem Solving Website, The Mathematical Association of America, Entry 2022 AMC12/AHSME

Cf. A001008/A002805 (harmonic numbers), A003418 (LCM).
Cf. A110566 (common factor).
Cf. A098464 (complement), A112813.
Cf. A330680 (numbers that begin a run of consecutive integers not in the sequence).

Select[Range[100], Denominator[HarmonicNumber[#]] < LCM @@ Range[#] &] (* Amiram Eldar, Nov 25 2022 *)

isok(n) = lcm(vector(n, i, i)) <> denominator(sum(i=1, n, 1/i)); \\ Thomas Scheuerle, Nov 23 2022

A110566(a(n)) > 1. - Thomas Scheuerle, Nov 23 2022

cp's OEIS Frontend

A112814 Numbers k such that lcm(1,2,3,...,k)/5 equals the denominator of the k-th harmonic number H(k).

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A112815 Numbers k such that lcm(1,2,3,...,k)/7 equals the denominator of the k-th harmonic number H(k).

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A112816 Numbers k such that lcm(1,2,3,...,k)/9 equals the denominator of the k-th harmonic number H(k).

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A112817 Numbers k such that lcm(1,2,3,...,k)/11 equals the denominator of the k-th harmonic number H(k).

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A112818 Numbers k such that lcm(1,2,3,...,k)/13 equals the denominator of the k-th harmonic number H(k).

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A112819 Numbers k such that lcm(1,2,3,...,k)/15 equals the denominator of the k-th harmonic number H(k).

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A112820 Numbers k such that lcm(1,2,3,...,k)/17 equals the denominator of the k-th harmonic number H(k).

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A112821 Numbers n such that lcm(1,2,3,...,n)/19 equals the denominator of the n-th harmonic number H(n).

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