A003418 -id:A003418 - cp's OEIS frontend

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

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)).

A096180 Triangle read by rows: fraction of integers having k of the first n positive integers as divisors is T(n,k)/A003418(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 4, 3, 1, 16, 20, 16, 7, 1, 16, 20, 12, 6, 5, 1, 96, 136, 92, 48, 36, 11, 1, 192, 272, 136, 124, 66, 38, 11, 1, 576, 720, 464, 360, 206, 122, 58, 13, 1, 576, 720, 392, 384, 188, 154, 70, 26, 9, 1, 5760, 7776, 4640, 4232, 2264, 1728, 854, 330, 116, 19

Offset: 1

Matthew Vandermast, Jun 19 2004

Sum of entries in n-th row is A003418(n), the least common multiple of integers 1 to n.

			Triangle begins:
1
1 1
2 3 1
4 4 3 1
16 20 16 7 1
16 20 12 6 5 1

Cf. A003418, A217863 (first column).

lcmn(n) = lcm(vector(n, k, k));
rowd(n) = {v = vector(n); for (k=1, lcmn(n), d = divisors(k); v[sum(j=1, #d, d[j]<=n)]++;); v;} \\ Michel Marcus, Apr 29 2017

A139547 Triangle read by rows: T(n,k) = A003418(A010766).

Original entry on oeis.org

1, 2, 1, 6, 1, 1, 12, 2, 1, 1, 60, 2, 1, 1, 1, 60, 6, 2, 1, 1, 1, 420, 6, 2, 1, 1, 1, 1, 840, 12, 2, 2, 1, 1, 1, 1, 2520, 12, 6, 2, 1, 1, 1, 1, 1, 2520, 60, 6, 2, 2, 1, 1, 1, 1, 1, 27720, 60, 6, 2, 2, 1, 1, 1, 1, 1, 1, 27720, 60, 12, 6, 2, 2, 1, 1, 1, 1, 1, 1, 360360, 60, 12, 6, 2, 2, 1, 1, 1, 1, 1

Offset: 0

Mats Granvik, Apr 27 2008, May 07 2008

This triangle fits the formula of I. Vardi in the Mathworld link about the von Mangoldt function. That formula is the basis for Chebyshev's estimate for the number of primes.

			Triangle begins:
1;
2,1;
6,1,1;
12,2,1,1;
60,2,1,1,1;
60,6,2,1,1,1;
420,6,2,1,1,1,1;
840,12,2,2,1,1,1,1;
2520,12,6,2,1,1,1,1,1;
2520,60,6,2,2,1,1,1,1,1;
27720,60,6,2,2,1,1,1,1,1,1;
27720,60,12,6,2,2,1,1,1,1,1,1;
360360,60,12,6,2,2,1,1,1,1,1,1,1;
...

I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 155.

Eric Weisstein's World of Mathematics, Mangoldt Function..

Cf. A000142, A010766, A014963, A003418, A139550, A139552, A139554.

nn = 13; a = Exp[Accumulate[MangoldtLambda[Range[nn]]]]; Flatten[Table[Table[a[[Floor[n/k]]], {k, 1, n}], {n, 1, nn}]][[1 ;; 89]]
(*As a limit of a recurrence*)
Clear[t, s, n, k, z, nn, ss, a, aa];(*z=1 corresponds to Zeta[1],z=2 corresponds to Zeta[2],z=ZetaZero[1] corresponds to Zeta[ZetaZero[1]],etc.*) z = 1; a = Normal[Series[Zeta[s], {s, z, 0}]]; ss = 10^40; s = N[z + 1/ss, 10^2]; nn = 13; t[n_, k_] := t[n, k] = If[k == 1, n*Zeta[s] - Sum[t[n, i]/i^(s - 1), {i, 2, n}], If[n >= k, t[Floor[n/k], 1], 0], 0]; aa = Table[Table[If[n >= k, t[n, k] - a, 0], {k, 1, n}], {n, 1, nn}]; Flatten[Round[Exp[aa]]][[1 ;; 89]]
(* Mats Granvik, Jun 05 2016 *)

From Mats Granvik, Jun 05 2016: (Start)
T(n,k)=A003418(floor(n/k)).
Recurrence involving log(n!):
Let s=1.
T(n, k) = if k = 1 then log(n!) - Sum_{i=2..n} T(n, i)/i^(s - 1) else if n >= k then T(floor(n/k), 1) else 0 else 0.
Recurrence involving the Riemann zeta function:
Let z = 1.
Let a = the series expansion of zeta(s) at z.
Let ss -> Infinity.
Let s = z + 1/ss.
Then T(n,k) is generated by the recurrence:
a + Ts(n, k) = if k = 1 then n*zeta(s) - Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n >= k then Ts(floor(n/k), 1) else 0 else 0.
(End)

Edited by Mats Granvik, Jun 28 2009

Further edits from N. J. A. Sloane, Jul 03 2009

A236856 Partial sums of A003418 starting summing from A003418(1), with a(0) = 0.

Original entry on oeis.org

0, 1, 3, 9, 21, 81, 141, 561, 1401, 3921, 6441, 34161, 61881, 422241, 782601, 1142961, 1863681, 14115921, 26368161, 259160721, 491953281, 724745841, 957538401, 6311767281, 11665996161, 38437140561, 65208284961, 145521718161, 225835151361, 2554924714161

Offset: 0

Antti Karttunen, Feb 27 2014

nonn

Similar comments about the trailing digits apply here as in A173185.
a(n) gives the position of the last element of row n in irregular tables like A238280.
From a(2)=3 onward all terms are divisible by three.
a(n) is divisible by 73 for n >= 72. Therefore a(n)/3 is prime for only 13 values of n: 3, 4, 6, 8, 9, 12, 16, 22, 23, 31, 35, 48 and 53. - Amiram Eldar, Sep 19 2022

Antti Karttunen, Table of n, a(n) for n = 0..250

One less than A173185.

Cf. A003418, A007489, A231722, A236857, A236858.

Prepend[Accumulate @ Table[LCM @@ Range[n], {n, 1, 30}], 0] (* Amiram Eldar, Sep 19 2022 *)

(define (A236856 n) (if (< n 2) n (+ (A236856 (- n 1)) (A003418 n))))

a(n) = A173185(n)-1.

A236857 Each n occurs the least common multiple (LCM) of {1, 2, ..., n} (= A003418(n)) times.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Feb 27 2014

Least k such that A236856(k) >= n.
Zero occurs once at a(0), because A003418(0)=1 by definition.
Useful when computing irregular tables like A238280, as a(n) gives the row index of the n-th term in such sequences. Note that as A238280 begins with row 1, it starts referring to this sequence only from a(1)=1 onward.

			Can be viewed as an irregular table, where each row n (starting from row zero) contains A003418(n) copies of n:
0;
1;
2, 2;
3, 3, 3, 3, 3, 3;
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;
...

Antti Karttunen, Rows 0..10 of the table, flattened

Cf. A003418, A084556, A236856, A236858.

Join[{0},Flatten[Table[PadRight[{},LCM@@Range[n],{n}],{n,6}]]] (* Harvey P. Dale, Jul 29 2021 *)

A059794 a(n) = n* - 2^(n-1), where n* (A003418) = least common multiple of the numbers [1,...,n].

Original entry on oeis.org

0, 0, 2, 4, 44, 28, 356, 712, 2264, 2008, 26696, 25672, 356264, 352168, 343976, 687952, 12186704, 12121168, 232530416, 232268272, 231743984, 230695408, 5350034576, 5345840272, 26754367184, 26737589968, 80246324336, 80179215472

Offset: 1

Kathleen Cussen (ehlana52(AT)hotmail.com), Feb 22 2001

It is known that this sequence is always nonnegative - see references.

LCM(1,2,3...n) = n* LCM( binomial(n-1,0), binomial(n-1,1),..., binomial(n-1,n-1)) - see American Mathematical Monthly E2686. - Paul Mills, Feb 14 2002

			Let n=4. Then n*=12 and 2^(4-1)=8. Then we calculate 12-8=4 to be the second term of the sequence.

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, pp. 12-13, Publications de l'Institut Cartan, 1990.

T. D. Noe, Table of n, a(n) for n=1..200
Peter L. Montgomery, LCM of Binomial Coefficients, Problem E2686, American Mathematical Monthly, Vol. 86 (1979), p. 131.
M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982) 126-129.
M. Nair, A new method in elementary prime number theory, J. London Math. Soc. 25 (1982) 385-391.

Cf. A003418, A067068, A068510, A068511.

A059794 := n->lcm(seq(i,i=1..n))-2^(n-1);

a[n_] := LCM @@ Range[n] - 2^(n-1); Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jul 05 2012 *)

a(n) = lcm(vector(n, i, i)) - 2^(n-1); \\ Michel Marcus, Jan 26 2015

Corrected and extended by Vladeta Jovovic, Feb 24 2001

References from Jean-Paul Allouche, Feb 17 2002

A064858 Denominators of partial sums of reciprocals of lcm(1..n) = A003418(n).

Original entry on oeis.org

1, 2, 3, 4, 30, 60, 14, 840, 315, 504, 6930, 9240, 180180, 360360, 30030, 720720, 2042040, 2450448, 10581480, 1989680, 58198140, 232792560, 1338557220, 254963280, 13385572200, 26771144400, 5736673800, 6177956400, 291136195350

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) = 14.

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

Cf. A003418, A064857, 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(denom, P); # Robert Israel, Jan 30 2025

q[x_] := Apply[LCM, Table[j, {j, 1, x}]] Table[Denominator[Apply[Plus, Table[1/q[w], {w, 1, m}]]], {m, 1, 30}]

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

A225637 a(n) = A003418(n)/A225629(n).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 2, 5, 7, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Antti Karttunen, May 13 2013

nonn

For n >= 2, a(n) is the final factor by which the A225629(n) needs to be multiplied that it finally reaches the fixed point A003418(n) of the column n of A225630.
The first composite, 4, occurs at n=20. The first composite which is not power of prime, namely 6, occurs at n=61.
For all n >= 3, a(n) divides A225558(n).

Cf. A003418, A225629, A225636, A225558, A225652, A225656.

(define (A225637 n) (/ (A003418 n) (A225629 n)))

A225643 Number of steps to reach a fixed point (A003418(n)), when starting from partition {n} of n and continuing with the process described in A225642.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 4, 4, 4, 5, 4, 6, 5, 6, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 8, 9, 8, 8, 9, 9, 9, 9, 9, 10, 11, 10, 10, 11, 11, 10, 10, 12, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 11, 13, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14

Offset: 0

Antti Karttunen, May 15 2013

nonn

a(0)=0, as its only partition is an empty partition {}, and by convention lcm()=1, thus it takes no steps to reach from 1 to A003418(0)=1.

The records occur at positions 0, 3, 5, 9, 11, 13, 19, 27, 30, 33, 43, 44, 51, 65, 74, 82, ... and they seem to occur in order, i.e., as A001477. Thus the record-positions probably also give the left inverse function for this sequence. It also seems that each integer occurs only finite times in this sequence, so there should be a right inverse function as well.

Cf. A225644, A225633, A225639, A226055, A226056.

Scheme
```
(define (A225643 n) (-1+ (A225644 n)))
```

a(n) = A225644(n) - 1.

A250270 Products of terms of A003418.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 960, 1024, 1152, 1296, 1440, 1536, 1680, 1728, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456

Offset: 1

Matthew Vandermast, Dec 16 2014

nonn

Includes all factorials and Jordan-Polya numbers, since n! = Product_{i = 1..n} A003418(floor(n/i)) for positive n.

			720 = 2*6*60 = 12*60. Since 2, 6, 12 and 60 are all terms of A003418, 720 is a term of this sequence.

Michel Marcus, Table of n, a(n) for n = 1..5000

Range of values of A253139. Subsequences include A000142, A001013, A001813, A025527, A064350, A166338, A250569.

Subsequence of A025487.

f(n) = lcm(vector(n, i, i)); \\ A003418
mul(x,y) = x*y;
lista(nn) = {my(v = vector(nn, k, f(k)), lim = f(nn+1), ok = 0, nv); while (!ok,  nv = select(x->(xMichel Marcus, May 09 2021

cp's OEIS Frontend

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

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A096180 Triangle read by rows: fraction of integers having k of the first n positive integers as divisors is T(n,k)/A003418(n).

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A139547 Triangle read by rows: T(n,k) = A003418(A010766).

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A236856 Partial sums of A003418 starting summing from A003418(1), with a(0) = 0.

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A236857 Each n occurs the least common multiple (LCM) of {1, 2, ..., n} (= A003418(n)) times.

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A059794 a(n) = n* - 2^(n-1), where n* (A003418) = least common multiple of the numbers [1,...,n].

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A064858 Denominators of partial sums of reciprocals of lcm(1..n) = A003418(n).

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