A064858 -id:A064858 - cp's OEIS frontend

A064859 Decimal expansion of sum of reciprocals of lcm(1..n) = A003418(n).

1, 7, 8, 7, 7, 8, 0, 4, 5, 6, 1, 7, 2, 4, 6, 6, 5, 4, 6, 0, 6, 4, 9, 3, 4, 3, 2, 6, 0, 2, 5, 6, 6, 2, 7, 9, 4, 5, 9, 3, 9, 6, 1, 7, 4, 7, 2, 9, 6, 9, 6, 0, 8, 3, 7, 2, 5, 3, 0, 2, 6, 9, 9, 2, 9, 2, 2, 8, 9, 0, 2, 3, 5, 0, 8, 2, 2, 3, 2, 6, 1, 5, 5, 2, 8, 3, 3, 6, 8, 7, 8, 0, 8, 5, 6, 9, 7, 9, 7, 9, 9, 4, 6, 9, 5

Offset: 1

Labos Elemer, Oct 08 2001

This constant is irrational (Erdős and Graham, 1980). - Amiram Eldar, Apr 13 2020

			1.7877804561724665460649343260256627945939617472969608372530269929228902350...

Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980. See p. 65.
Martin Griffiths and Des MacHale, 99.04 Another irrational number, The Mathematical Gazette, Vol. 99, No. 544 (2015), pp. 130-133.
Michael Penn, An irrational sum, YouTube video, 2022.

Cf. A003418, A064857, A064858.

f[n_] := LCM @@ Range@ n; RealDigits[Plus @@ (1/Array[f, 255]), 10, 111][[1]] (* Robert G. Wilson v, Jul 11 2011 *)

suminf(k=1, 1/lcm(vector(k, j, j))) \\ Michel Marcus, Mar 11 2018

Equals Sum_{j>=1} 1/lcm(1..j).

A064857 Numerators of partial sums of reciprocals of lcm(1..n) = A003418(n).

Original entry on oeis.org

1, 3, 5, 7, 53, 107, 25, 1501, 563, 901, 12389, 16519, 322121, 644243, 53687, 1288489, 3650719, 4380863, 18917363, 3557111, 104045497, 416181989, 2393046437, 455818369, 23930464373, 47860928747, 10255913303, 11044829711

Offset: 1

Labos Elemer, Oct 08 2001

			n = 7: LCM values: 1, 2, 6, 12, 60, 60, 420; partial sum = 1 + 1/2 + 1/6 + 1/12 + 1/60 + 1/60 + 1/420 = (420 + 210 + 70 + 35 + 7 + 7 + 1)/420 = 750/420 = 25/14, so a(7) = 25.

Robert Israel, Table of n, a(n) for n = 1..2296

Cf. A003418, A064858, A064859.

R:= 1: m:= 1: for n from 2 to 100 do m:= ilcm(m,n); R:= R,1/m od:
P:= ListTools:-PartialSums([R]):
map(numer,P); # Robert Israel, Jan 30 2025

q[x_] := Apply[LCM, Table[j, {j, 1, x}]] Table[Numerator[Apply[Plus, Table[1/q[w], {w, 1, m}]]], {m, 1, 30}]
Accumulate[Table[1/LCM@@Range[n],{n,30}]]//Numerator (* Harvey P. Dale, Aug 08 2021 *)

a(n) = numerator(Sum_{j=1..n} 1/lcm(1..n)).

A064889 Denominators of partial sums of reciprocals of A051451 (A051451 includes lcm(1,...,x), x=power of prime from A000961 and also contains 1).

Original entry on oeis.org

1, 2, 3, 4, 30, 420, 840, 1260, 9240, 9009, 240240, 471240, 232792560, 535422888, 26771144400, 181704600, 4822131600, 36100888223400, 3702655202400, 76327592243760, 737576396429600, 362291852261631600, 6416241209619040800

Offset: 1

Labos Elemer, Oct 11 2001

			n=6, sum = (1/1) + (1/2) + (1/6) + (1/12) + (1/60) + (1/420) = 743/420, the denominator = a(6) = 743. Remark: If 1 is omitted from A051451, the denominators of partial sums of reciprocals do not change; however, numerators are changing, not compatible with these denominators (see A064888).

Cf. A051451, A003418, A000961, A064857, A064858, A064859, A064888, A064890.

a(n) = denominator(Sum_{k=1..n} 1/A051451(k)).

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A064859 Decimal expansion of sum of reciprocals of lcm(1..n) = A003418(n).

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A064857 Numerators of partial sums of reciprocals of lcm(1..n) = A003418(n).

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A064889 Denominators of partial sums of reciprocals of A051451 (A051451 includes lcm(1,...,x), x=power of prime from A000961 and also contains 1).

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