A110566 -id:A110566 - cp's OEIS frontend

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

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

A112811 Terms of A110566 grouped.

Original entry on oeis.org

1, 3, 1, 3, 15, 45, 15, 3, 1, 11, 77, 7, 1, 3, 9, 27, 9, 3, 33, 11, 1, 25, 5, 55, 275, 25, 1, 13, 39, 3, 9, 27, 9, 3, 1, 17, 1, 49, 7, 49, 931, 19, 1, 11, 319, 11, 319, 11, 1, 3, 75, 1725, 345, 15, 645, 1935, 5805, 29025, 675, 41175, 13725, 549, 20313, 6771, 183, 3, 411, 15207

Offset: 1

Robert G. Wilson v, Sep 20 2005

nonn

A110566: LCM{1,2,...,n}/denominator of harmonic number H(n).

The factor of change from a(n) to a(n+1) is: 3,3,3,5,3,3,5,3,11,7,11,7,3,3,3,3,3,11,3,11,25,5,11,5,11,25,13,3,13,3,3,3,3,..., . see A110268.

Cf. A110566, A112812, A110268.

f[n_] := LCM @@ Range[n]/Denominator[HarmonicNumber[n]]; Flatten[Union /@ Split[Table[f[n], {n, 703}]]]

A110268 Consider the sequence A110566: lcm{1,2,...,n}/denominator of harmonic number H(n). a(n) is the factor that is changed going from A110566(n) to A110566(n+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 11, 1, 1, 1, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 11, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

a(n) is always an odd prime power, A061345.

			A110566(4) through A110566(10) are {1,1,3,3,3,1,1}, therefore the factors are 1,3,1,1,3,1.

Cf. A110566, A112811.

f[n_] := LCM @@ Range[n]/Denominator[HarmonicNumber[n]]; Table[ LCM[f[n], f[n + 1]]/GCD[f[n], f[n + 1]], {n, 104}]

f(n) = lcm(vector(n, k, k))/denominator(sum(k=1, n, 1/k));
a(n) = my(x = f(n+1)/f(n)); if (x > 1, x, 1/x); \\ Michel Marcus, Mar 07 2018

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

A112812 a(n) is length of n-th run in A110566.

Original entry on oeis.org

5, 3, 9, 2, 1, 3, 1, 2, 6, 9, 2, 5, 5, 9, 3, 3, 3, 5, 4, 7, 12, 5, 5, 10, 1, 4, 31, 6, 7, 20, 9, 9, 9, 27, 29, 17, 5, 7, 35, 6, 1, 18, 2, 14, 29, 29, 29, 20, 2, 14, 6, 19, 4, 30, 8, 27, 6, 2, 8, 11, 4, 4, 19, 18, 5, 14, 18, 26, 11, 72, 10, 19, 6, 11, 22, 11, 33, 6, 5, 22, 4, 7, 99, 97, 2, 44, 9

Offset: 1

Robert G. Wilson v, Sep 20 2005

nonn

Cf. A110566, A112811.

f[n_] := LCM @@ Range[n]/Denominator[HarmonicNumber[n]]; Length /@ Split[Table[f[n], {n, 1220}]]

Edited by Don Reble, Nov 08 2005

Definition clarified by Omar E. Pol, Dec 26 2008

A338120 a(n) is the least index k such that the n-th odd squarefree number A056911(n) divides A110566(k).

Original entry on oeis.org

1, 6, 20, 42, 33, 156, 20, 272, 342, 2058, 506, 377, 930, 77, 14406, 629, 162, 1640, 559, 2162, 4624, 1166, 110, 6498, 3422, 610, 342732, 4422, 506, 4970, 5256, 42, 6162, 6806

Offset: 1

Amiram Eldar, Jan 29 2021

According to a theorem proven by Shiu (2016), a(n) exists for all n.

			   n  A056911(n)  a(n) = k          A110566(k)
  --  ----------  --------  --------------------------
   1       1            1            1 =  1 * 1
   2       3            6            3 =  3 * 1
   3       5           20           15 =  5 * 3
   4       7           42           77 =  7 * 11
   5      11           33           11 = 11 * 1
   6      13          156           13 = 13 * 1
   7      15           20           15 = 15 * 1
   8      17          272           17 = 17 * 1
   9      19          342          931 = 19 * 49
  10      21         2058         1911 = 21 * 91
  11      23          506         1725 = 23 * 75
  12      29          377          319 = 29 * 11
  13      31          930         3751 = 31 * 121
  14      33           77           33 = 33 * 1
  15      35        14406   2430488445 = 35 * 69442527
  16      37          629        20313 = 37 * 549
  17      39          162           39 = 39 * 1
  18      41         1640         6519 = 41 * 159
  19      43          559          645 = 43 * 15
  20      47         2162        12831 = 47 * 273
  21      51         4624         9537 = 51 * 187
  22      53         1166           53 = 53 * 1
  23      55          110           55 = 55 * 1
  24      57         6498    419498967 = 57 * 7359631
  25      59         3422         6431 = 59 * 109
  26      61          610        41175 = 61 * 675
  27      65       342732       974285 = 65 * 14989
  28      67         4422         2211 = 67 * 33
  29      69          506         1725 = 69 * 25
  30      71         4970         2343 = 71 * 33
  31      73         5256         7227 = 73 * 99
  32      77           42           77 = 77 * 1
  33      79         6162     91801713 = 79 * 1162047
  34      83         6806      1200097 = 83 * 14459

Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.

Cf. A002805, A003418, A110566, A056911, A112822.

max = 64; osf = Select[Range[1, 64, 2], SquareFreeQ]; m = Length[osf]; c = 0; s = Table[0, {m}]; h = 0; lcm = 1; n = 1; While[c < m, h += 1/n; lcm = LCM[lcm, n];  r = lcm/Denominator[h]; Do[If[s[[k]] == 0 && Divisible[r, osf[[k]]], c++; s[[k]] = n], {k, 1, m}]; n++]; s

A025529 a(n) = (1/1 + 1/2 + ... + 1/n)*lcm{1,2,...,n}.

Original entry on oeis.org

1, 3, 11, 25, 137, 147, 1089, 2283, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 42822903, 825887397, 837527025, 848612385, 859193865, 19994251455, 20217344325, 102157567401, 103187226801, 312536252003, 315404588903, 9227046511387

Offset: 1

Clark Kimberling

nonn

First column of A027446. - Eric Desbiaux, Mar 29 2013
From Amiram Eldar and Thomas Ordowski, Aug 07 2019: (Start)
By Wolstenholme's theorem, if p > 3 is a prime, then p^2 | a(p-1).
Conjecture: for n > 3, if n^2 | a(n-1), then n is a prime.
Note that if n = p^2 with prime p > 3, then n | a(n-1).
It seems that composite numbers n such that n | a(n-1) are only the squares n = p^2 of primes p > 3.
Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164.
The n-th triangular number n(n+1)/2 | a(n) for n = 1, 2, 6, 4422, ... (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Frank A. Haight, and Robert B. Jones., "A probabilistic treatment of qualitative data with special reference to word association tests." Journal of Mathematical Psychology 11.3 (1974): 237-244. [Denominators of fractions in Eq. 21.] [Annotated scanned copy]
Frank A. Haight and N. J. A. Sloane, Correspondence, 1975
Yilmaz Simsek, Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions, arXiv:2108.10756 [math.NT], 2021.

Differs from A096617 at 7th term.

Cf. A001008, A002805, A027446, A110566.

List([1..30],n->Sum([1..n],k->1/k)*Lcm([1..n])); # Muniru A Asiru, Apr 02 2018

[HarmonicNumber(n)*Lcm([1..n]):n in [1..30]]; // Marius A. Burtea, Aug 07 2019

a:= n-> add(1/k, k=1..n)*ilcm($1..n):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 14 2013

Table[HarmonicNumber[n]*LCM @@ Range[n], {n, 27}] (* Arkadiusz Wesolowski, Mar 29 2012 *)

a(n) = sum(k=1, n, 1/k)*lcm([1..n]); \\ Michel Marcus, Apr 02 2018

a(n) = A001008(n)*A110566(n). - Arkadiusz Wesolowski, Mar 29 2012

a(n) = Sum_{k=1..n} lcm(1,2,...,n)/k. - Thomas Ordowski, Aug 07 2019

A098464 Numbers k such that lcm(1,2,3,...,k) equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149

Offset: 1

T. D. Noe, Sep 09 2004

Numbers k such that A110566(k) = 1.

Shiu (2016) conjectured that this sequence is infinite. - Amiram Eldar, Feb 02 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A002805 (denominator of H(n)), A003418 (lcm(1, 2, ..., n)), A110566.

Select[Range[250], LCM@@Range[ # ]==Denominator[HarmonicNumber[ # ]]&]

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

from fractions import Fraction
from sympy import lcm
k, l, h, A098464_list = 1, 1, Fraction(1, 1), []
while k < 10**6:
    if l == h.denominator:
        A098464_list.append(k)
    k += 1
    l = lcm(l,k)
    h += Fraction(1,k) # Chai Wah Wu, Mar 07 2021

A112822 Least number k such that lcm{1,2,...,k}/denominator of harmonic number H(k) = 2n-1.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 15 2005

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

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

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

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

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

A112813 Numbers k such that lcm(1,2,3,...,k)/3 equals the denominator of the k-th harmonic number H(k).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

When 3 occurs in A110566.

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

Cf. A002805, A003418, A110566.

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

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

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

cp's OEIS Frontend

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

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A112811 Terms of A110566 grouped.

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A110268 Consider the sequence A110566: lcm{1,2,...,n}/denominator of harmonic number H(n). a(n) is the factor that is changed going from A110566(n) to A110566(n+1).

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A112809 Positions of records in A110566.

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A112812 a(n) is length of n-th run in A110566.

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A338120 a(n) is the least index k such that the n-th odd squarefree number A056911(n) divides A110566(k).

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A025529 a(n) = (1/1 + 1/2 + ... + 1/n)*lcm{1,2,...,n}.

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A098464 Numbers k such that lcm(1,2,3,...,k) equals the denominator of the k-th harmonic number H(k).

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