A098464 -id:A098464 - cp's OEIS frontend

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

A112810 Records in A110566 (lcm{1,2,...,n}/denominator of harmonic number H(n)).

Original entry on oeis.org

1, 3, 15, 45, 77, 275, 931, 1725, 1935, 5805, 29025, 41175, 166803, 1039533, 1162047, 91801713, 419498967, 2183383175, 19691916585, 216611082435, 2382721906785, 113804487945521, 22211221792244703, 422013214052649357, 425137351586922079, 936039253001457601

Offset: 1

Robert G. Wilson v, Sep 19 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..28

Cf. A110566, A001008, A002805, A003418, A025529, A098464, A112809.

c = 0; a = h = 1; t = {}; Do[a = LCM[a, n]; h = h + 1/n; b = a/Denominator[h]; If[b > c, c = b; AppendTo[t, b]], {n, 10^6}]; t

lista(nn) = {rec = 0; for (n=1, nn, new = lcm(vector(n, k, k))/denominator(sum(k=1, n, 1/k)); if (new > rec, print1(new, ", "); rec = new););} \\ Michel Marcus, Mar 07 2018

a(n) = A110566(A112809(n)).

a(25)-a(26) from Max Alekseyev, Nov 29 2013

A330680 Numbers that begin a run of consecutive integers k such that the denominator of the k-th harmonic number is lcm(1..k).

Original entry on oeis.org

1, 9, 27, 49, 88, 125, 243, 289, 361, 484, 841, 968, 1164, 1331, 1369, 2401, 3125, 3488, 3721, 6889, 7085, 7761, 7921, 8342, 8502, 9156, 10648, 19683, 22208, 22801, 25886, 28561, 29929, 30877, 32041, 32761, 33178, 36481, 59049, 83521, 87079, 88307, 92199

Offset: 1

Jon E. Schoenfield, Dec 24 2019

nonn

A098464 lists the numbers k such that lcm(1,2,3,...,k) equals the denominator of the k-th harmonic number H(k) = 1/1 + 1/2 + 1/3 + ... + 1/k.

			The numbers k such that the denominator of the k-th harmonic number equals lcm(1..k) begin with the following runs of consecutive integers:
    1,   2,   3,   4,   5;
    9,  10,  11,  12,  13,  14,  15,  16,  17;
   27,  28,  29,  30,  31,  32;
   49,  50,  51,  52,  53;
   88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,  99;
  125, 126, 127, ...
so this sequence begins 1, 9, 27, 49, 88, 125, ...

Chai Wah Wu, Table of n, a(n) for n = 1..308

Cf. A002805 (denominator of H(n)), A003418 (lcm(1..n)), A098464 (numbers k such that A002805(k)=A003418(k)).

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

A112809 Positions of records in A110566.

Original entry on oeis.org

1, 6, 20, 21, 42, 120, 342, 506, 567, 594, 600, 610, 2184, 4896, 6108, 6162, 6498, 12760, 14067, 14157, 14201, 93942, 123462, 123519, 734413, 2451397, 4591010, 11571129, 13346540, 13619348, 13619790, 46180567

Offset: 1

Robert G. Wilson v, Sep 19 2005

Cf. A110566, A001008, A002805, A003418, A025529, A098464, A112810.

c = 0; a = h = 1; t = {}; Do[a = LCM[a, n]; h = h + 1/n; b = a/Denominator[h]; If[b > c, c = b; AppendTo[t, n]], {n, 10^6}]; t

lista(nn) = {rec = 0; for (n=1, nn, new = lcm(vector(n, k, k))/denominator(sum(k=1, n, 1/k)); if (new > rec, print1(n, ", "); rec = new););} \\ Michel Marcus, Mar 07 2018

a(27)-a(28) from Amiram Eldar, Dec 18 2018
a(29)-a(31) from Chai Wah Wu, Mar 08 2021
a(32) from Chai Wah Wu, Mar 14 2021

A342351 Numbers k such that lcm(1,2,3,...,k)/23 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

11881, 11882, 11883, 11884, 11885, 11886, 11887, 11888, 11889, 11890, 11891, 11892, 11893, 11894, 11895, 11896, 11897, 11898, 11899, 11900, 11901, 11902, 11903, 11904, 11905, 11906, 11907, 11908, 11909, 11910, 11911, 11912, 11913, 11914, 11915, 11916, 11917

Offset: 1

Chai Wah Wu, Mar 17 2021

nonn

Positions where 23 occurs in A110566.

Chai Wah Wu, Table of n, a(n) for n = 1..9014
Tanya Khovanova, Non Recursions

Cf. A002805, A003418, A110566.

Cf. A098464, A112813, A112814, A112815, A112816, A112817, A112818, A112819, A112820, A112821, A112822, A342350.

cp's OEIS Frontend

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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A112810 Records in A110566 (lcm{1,2,...,n}/denominator of harmonic number H(n)).

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A330680 Numbers that begin a run of consecutive integers k such that the denominator of the k-th harmonic number is lcm(1..k).

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

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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).

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A112809 Positions of records in A110566.

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A342351 Numbers k such that lcm(1,2,3,...,k)/23 equals the denominator of the k-th harmonic number H(k).

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