A110566 -id:A110566

A112810

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

+20
3

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

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

OFFSET

1,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 19 2005

EXTENSIONS

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

STATUS

approved

A112811

Terms of A110566 grouped.

+20
2

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

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

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..68.

MATHEMATICA

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

CROSSREFS

Cf. A110566, A112812, A110268.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 20 2005

STATUS

approved

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).

+20
1

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

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

OFFSET

1,5

COMMENTS

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

LINKS

Table of n, a(n) for n=1..104.

EXAMPLE

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

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

Cf. A110566, A112811.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 17 2005

STATUS

approved

A112809

Positions of records in A110566.

+20
1

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

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..32.

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

more,nonn

AUTHOR

Robert G. Wilson v, Sep 19 2005

EXTENSIONS

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

STATUS

approved

A112812

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

+20
1

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

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

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..87.

MATHEMATICA

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

CROSSREFS

Cf. A110566, A112811.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Sep 20 2005

EXTENSIONS

Edited by Don Reble, Nov 08 2005

Definition clarified by Omar E. Pol, Dec 26 2008

STATUS

approved

A338120

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

+20
0

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

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

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..34.

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

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Jan 29 2021

STATUS

approved

A025529

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

+10
18

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

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

OFFSET

1,2

COMMENTS

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)

LINKS

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.

FORMULA

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

MAPLE

a:= n-> add(1/k, k=1..n)*ilcm($1..n):

seq(a(n), n=1..30); # Alois P. Heinz, Mar 14 2013

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

Differs from A096617 at 7th term.

Cf. A001008, A002805, A027446, A110566.

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

A098464

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

+10
18

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

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

OFFSET

1,2

COMMENTS

Numbers k such that A110566(k) = 1.

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

LINKS

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.

MATHEMATICA

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

PROG

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

(Python)

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

T. D. Noe, Sep 09 2004

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