A003418 -id:A003418 - cp's OEIS frontend

A180000 a(n) = lcm{1,2,...,n} / swinging_factorial(n) = A003418(n) / A056040(n).

1, 1, 1, 1, 2, 2, 3, 3, 12, 4, 10, 10, 30, 30, 105, 7, 56, 56, 252, 252, 1260, 60, 330, 330, 1980, 396, 2574, 286, 2002, 2002, 15015, 15015, 240240, 7280, 61880, 1768, 15912, 15912, 151164, 3876, 38760, 38760, 406980, 406980, 4476780, 99484, 1144066

Offset: 0

Peter Luschny, Aug 17 2010

nonn

Characterization: Let e_{p}(m) denote the exponent of the prime p in the prime factorization of m and [.] denote the Iverson bracket, then
e_{p}(a(n)) = Sum_{k>=1} [floor(n/p^k) is even].
This implies, among other things, that no prime > floor(n/2) can divide a(n). The prime exponents e_{2}(a(2n)) give Guy Steele's sequence GS(5,3) A080100.
Asymptotics: log a(n) ~ n(1 - log 2). It is conjectured that log a(n) ~ n(1 - log 2) + O(n^{1/2+eps}) for all eps > 0.
Bounds: A056040(floor(n/3)) <= a(n) <= A056040(floor(n/2)) if n >= 285.

Peter Luschny, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A003418, A014963, A056040.

a := proc(n) local A014963, k;
A014963 := proc(n) if n < 2 then 1 else numtheory[factorset](n);
if 1 < nops(%) then 1 else op(%) fi fi end;
mul(A014963(k)*(k/2)^((-1)^k), k=1..n)/2^n end;
# Also:
A180000 := proc(n) local lcm, sf;
lcm := ilcm(seq(i,i=1..n));
sf := n!/iquo(n,2)!^2;
lcm/sf end;

a[0] = 1; a[n_] := LCM @@ Range[n] / (n! / Floor[n/2]!^2); Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Jul 23 2013 *)

L=1; X(n)={ ispower(n, , &n);if(isprime(n),n,1); }
Y(n)={ a=X(n); b=if(bitand(1,n),a,a*(n/2)^2); L=(b*L)/n; }
A180000_list(n)={ L=1; vector(n,m,Y(m)); }  \\ for n>0

def Exp(m,n) :
    s = 0; p = m; q = n//p
    while q > 0 :
        if is_even(q) :
            s = s + 1
        p = p * m
        q = n//p
    return s
def A180000(n) :
    A = [1,1,1,1,2,2,3,3,12]
    if n < 9 : return A[n]
    R = []; r = isqrt(n)
    P = Primes(); p = P.first()
    while p <= n//2 :
        if p <= r : R.append(p^Exp(p,n))
        elif p <= n//3 :
            if is_even(n//p) : R.append(p)
        else : R.append(p)
        p = P.next(p)
    return mul(x for x in R)

a(n) = 2^(-n)*Product_{1<=k<=n} A014963(k)*(k/2)^((-1)^k).

A049536 Primes of the form lcm(1, ..., n) + 1 = A003418(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 2521, 232792561, 26771144401, 72201776446801, 442720643463713815201, 718766754945489455304472257065075294401, 6676878045498705789701874602220118271269436344024536001

Offset: 1

Labos Elemer

nonn

			Lcm(9) + 1 = lcm(10) + 1 = 2521, a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..16

Subsequence of A070858.

Cf. A003418, A049537, A057824.

Select[Table[LCM@@Range[n]+1,{n,150}],PrimeQ]//Union (* Harvey P. Dale, May 31 2017 *)

N=1; print1(2); for(n=1,1e3, if(isprimepower(n,&p), N*=p; if(isprime(N+1), print1(", "N+1)))) \\ Charles R Greathouse IV, Nov 18 2015

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

Original entry on oeis.org

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).

A049537 Values of k for which A075059(k) = A003418(k) + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 19, 20, 21, 22, 25, 26, 31, 47, 48, 89, 90, 91, 92, 93, 94, 95, 96, 127, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 1369, 1370, 1371, 1372, 2251, 2252, 2253, 2254, 2255, 2256, 2257, 2258, 2259, 2260, 2261, 2262, 2263, 2264, 2265, 2266, 3271, 3272, 3273, 3274, 3275, 3276, 3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287

Offset: 1

Labos Elemer

nonn

			8 is not in the sequence because A075059(8) = 1 + A003418(8) = 1 + lcm(1, 2, ..., 8) = 841 = 29^2 is not prime.
127 is in the sequence because A075059(127) = 1 + A003418(127) = 1 + lcm(1, 2, ..., 127) = 6676878045498705789701874602220118271269436344024536001 is prime.

Jinyuan Wang, Table of n, a(n) for n = 1..150

Cf. A000040, A003418, A049536, A057825, A075059.

Join[{0}, Select[Range[250], PrimeQ[LCM@@Range[#]+1]&]] (* Harvey P. Dale, Nov 15 2011 *)

isok(n) = isprime(lcm(vector(n, i, i))+1); \\ Michel Marcus, Feb 25 2014

a(43)-a(57) from Ray Chandler, Jan 16 2009

a(1)=0 prepended and a(58)-a(86) added by Max Alekseyev, Sep 04 2015

A075059 a(n) = 1 + lcm(1, 2, ..., n) = 1 + A003418(n).

Original entry on oeis.org

2, 2, 3, 7, 13, 61, 61, 421, 841, 2521, 2521, 27721, 27721, 360361, 360361, 360361, 720721, 12252241, 12252241, 232792561, 232792561, 232792561, 232792561, 5354228881, 5354228881, 26771144401, 26771144401, 80313433201, 80313433201

Offset: 0

Amarnath Murthy, Sep 08 2002

nonn

Consider the triangle in which the n-th row contains the second run of n consecutive numbers such that the r-th term is divisible by r. Sequence gives the first column of the triangle. The first run trivially begins with 1.
Also the smallest of n consecutive integers (with the first greater than 1) divisible respectively by 1, 2, 3, ..., n. - Robert G. Wilson v, Oct 30 2014
Also the smallest number m > 1 such that m == 1 (mod i) for all 1 <= i <= n. - Franz Vrabec, Aug 18 2023

			First column of the triangle A075061:
   2;
   3,  4;
   7,  8,  9;
  13, 14, 15, 16;
  61, 62, 63, 64, 65;
  61, 62, 63, 64, 65, 66;
  ...

Cf. A003418, A060401, A065500, A070198, A072562, A075061, A249051.

[Exponent(SymmetricGroup(n))+1 : n in [1..30]]; // Vincenzo Librandi, Oct 31 2014

Table[LCM@@Range[n] + 1, {n, 30}] (* Harvey P. Dale, Nov 15 2011 *)

a(n)=lcm([1..n])+1 \\ Charles R Greathouse IV, Nov 04 2014

a(n) = 1 + A003418(n).

New definition from Vladeta Jovovic, Jun 16 2003
Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar
a(0)=2 prepended by Max Alekseyev, Sep 04 2015

A133232 Triangle T(n,k) read by rows with a minimum number of prime powers A100994 for which the least common multiple of T(n,1),..,T(n,n) is A003418(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 1, 3, 4, 5, 1, 1, 3, 4, 5, 1, 1, 1, 3, 4, 5, 1, 7, 1, 1, 3, 1, 5, 1, 7, 8, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 13, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1

Offset: 1

Mats Granvik, Oct 13 2007

Checked up to 28th row. The rest of the ones in the table are there for the least common multiple to calculate correctly.

			2 occurs 2*1 = 2 times in column 2.
3 occurs 3*2 = 6 times in column 3.
4 occurs 4*1 = 4 times in column 4.
5 occurs 5*4 = 20 times in column 5.
k occurs A133936(k) times in column k. The first rows of the triangle and the least common multiple of the rows are:
lcm{1} = 1
lcm{1, 2} = 2
lcm{1, 2, 3} = 6
lcm{1, 1, 3, 4} = 12
lcm{1, 1, 3, 4, 5} = 60
lcm{1, 1, 3, 4, 5, 1} = 60
lcm{1, 1, 3, 4, 5, 1, 7} = 420
lcm{1, 1, 3, 1, 5, 1, 7, 8} = 840
lcm{1, 1, 1, 1, 5, 1, 7, 8, 9} = 2520

Mats Granvik, Oct 13 2007, Table of n, a(n) for n = 1..406

Cf. A003418, A120112, A014963, A134579.

=if(and(row()>=column();row()A120112));column();1)

=if(and(n>=k; n < A014963*A100994); A100994; 1) - Mats Granvik, Jan 21 2008

A120112 := proc(n) 1-ilcm(seq(i,i=1..n+1))/ilcm(seq(i,i=1..n)) ; end proc:
A133232 := proc(n) if n < k*(1+abs(A120112(k-1))) then k else 1; end if; end proc:
seq(seq(A133232(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Nov 23 2010

b[n_] := b[n] = If[n == 0, 1, LCM @@ Range[n]];
c[n_] := 1 - b[n+1]/b[n];
T[n_, k_] := If[n < k*(1+Abs[c[k-1]]), k, 1];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2021 *)

T(n,k) = if nA120112(k-1)| then k, else 1 (1<=k<=n).

T(n,k) = if n < A014963(k)*A100994(k) then A100994(k), else 1 (1<=k<=n). - Mats Granvik, Jan 21 2008

Indices added to formulas by R. J. Mathar, Nov 23 2010

A225629 a(n) = Last value in column n of A225630 which is not yet the fixed point A003418(n) of that column.

Original entry on oeis.org

1, 1, 1, 3, 4, 30, 30, 84, 120, 1260, 840, 13860, 13860, 180180, 180180, 180180, 240240, 6126120, 6126120, 116396280, 58198140, 116396280, 116396280, 2677114440, 2677114440, 13385572200, 13385572200, 40156716600, 40156716600, 776363187600, 776363187600

Offset: 0

Antti Karttunen, May 13 2013

nonn

a(n) = also the second rightmost terms of row n of irregular table A225632.

a(0)= a(1) = 1 by convention.

Cf. A225630, A225637.

(define (A225629 n) (A225630bi n (max 0 (- (A225633 n) 1))))

a(n) = A225630(n,max(0,A225633(n)-1)).

A045948 a(n) = A003418(n)/A034386(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 12, 12, 12, 12, 12, 12, 12, 24, 24, 24, 24, 24, 24, 24, 24, 24, 120, 120, 360, 360, 360, 360, 360, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040

Offset: 1

Labos Elemer

nonn

			n=11: lcm(1..11) = 27720 = 8*9*5*7*11 = 2310*12. A034386(11)=2310, so the quotient is 12. Thus a(11) = 12.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A003418, A034386, A002110, A000142, A049536, A048148.

Table[Exp[Sum[MangoldtLambda[n], {n, 1, m}]]/ Product[x, {x, Prime[Range[PrimePi[m]]]}], {m, 1, 57}] (* Fred Daniel Kline, Apr 02 2015 *)

a(n)=lcm([1..n])/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Apr 02 2015; corrected by Michel Marcus, Dec 26 2020

A225558 a(n) = A003418(n)/A000793(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 10, 35, 56, 126, 84, 924, 462, 6006, 4290, 3432, 5148, 58344, 58344, 554268, 554268, 554268, 554268, 6374082, 6374082, 21246940, 21246940, 52151580, 34767720, 924241890, 504131940, 15628090140, 26447537160, 26447537160, 15628090140

Offset: 0

Antti Karttunen, May 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000793, A003418.

g:= proc(n) g(n):= `if`(n=0, 1, ilcm(n, g(n-1))) end:
b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, max(b(n, i-1),
           seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:= n->g(n)/b(n, `if`(n<8, 3,
    numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
seq(a(n), n=0..40); # Alois P. Heinz, May 22 2013

b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor }]]]]; a[0]=1; a[n_] := LCM @@ Range[n] / b[n, If[n<8, 3, PrimePi[ Ceiling[ 1.328*Sqrt[n*Log[n] // Floor]]]]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

a(n) = A003418(n)/A000793(n).

A074115(n)/a(n) = A025527(n).

A057825 Values of k for which A003418(k) - 1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 19, 20, 21, 22, 23, 24, 29, 30, 32, 33, 34, 35, 36, 47, 48, 61, 62, 63, 97, 98, 99, 100, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 233, 234, 235, 236, 237, 238, 307, 308, 309, 310, 401, 402, 403, 404, 405, 406, 407, 408, 887, 888, 889, 890

Offset: 1

Labos Elemer, Nov 08 2000

nonn

Fewer distinct primes than distinct values of a(n) are generated. So, e.g., k = 97, 98, 99, 100 all correspond to lcm([1..97]) - 1 = 69720375229712477164533808935312303556799, a prime.

Jinyuan Wang, Table of n, a(n) for n = 1..125

Cf. A003418, A049536, A049537, A051451.

isok(k) = ispseudoprime(lcm(vector(k, i, i))-1); \\ Jinyuan Wang, May 02 2020

cp's OEIS Frontend

A180000 a(n) = lcm{1,2,...,n} / swinging_factorial(n) = A003418(n) / A056040(n).

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A049536 Primes of the form lcm(1, ..., n) + 1 = A003418(n) + 1.

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

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A049537 Values of k for which A075059(k) = A003418(k) + 1 is prime.

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A075059 a(n) = 1 + lcm(1, 2, ..., n) = 1 + A003418(n).

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A133232 Triangle T(n,k) read by rows with a minimum number of prime powers A100994 for which the least common multiple of T(n,1),..,T(n,n) is A003418(n).

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A225629 a(n) = Last value in column n of A225630 which is not yet the fixed point A003418(n) of that column.

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