A096179 -id:A096179 - cp's OEIS frontend

A178938 Min{ n | A094348(n) element of {A096179(n,k) | 1 <= k <= n}}.

Original entry on oeis.org

1, 2, 4, 3, 4, 8, 18, 16, 5, 9, 8, 18, 16, 9, 7, 16, 15, 21, 16, 9, 16

Offset: 1

Peter Luschny, Dec 30 2010

nonn

Enrique Pérez Herrero, The triangle A096179.

A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 0

Roland Anderson (roland.anderson(AT)swipnet.se)

The minimal exponent of the symmetric group S_n, i.e., the least positive integer for which x^a(n)=1 for all x in S_n. - Franz Vrabec, Dec 28 2008
Product over all primes of highest power of prime less than or equal to n. a(0) = 1 by convention.
Also smallest number whose set of divisors contains an n-term arithmetic progression. - Reinhard Zumkeller, Dec 09 2002
An assertion equivalent to the Riemann hypothesis is: | log(a(n)) - n | < sqrt(n) * log(n)^2. - Lekraj Beedassy, Aug 27 2006. (This is wrong for n = 1 and n = 2. Should "for n large enough" be added? - Georgi Guninski, Oct 22 2011)
Corollary 3 of Farhi gives a proof that a(n) >= 2^(n-1). - Jonathan Vos Post, Jun 15 2009
Appears to be row products of the triangle T(n,k) = b(A010766) where b = A130087/A130086. - Mats Granvik, Jul 08 2009
Greg Martin (see link) proved that "the product of the Gamma function sampled over the set of all rational numbers in the open interval (0,1) whose denominator in lowest terms is at most n" equals (2*Pi)^(1/2)*a(n)^(-1/2). - Jonathan Vos Post, Jul 28 2009
a(n) = lcm(A188666(n), A188666(n)+1, ..., n). - Reinhard Zumkeller, Apr 25 2011
a(n+1) is the smallest integer such that all polynomials a(n+1)*(1^i + 2^i + ... + m^i) in m, for i=0,1,...,n, are polynomials with integer coefficients. - Vladimir Shevelev, Dec 23 2011
It appears that A020500(n) = a(n)/a(n-1). - Asher Auel, corrected by Bill McEachen, Apr 05 2024
n-th distinct value = A051451(n). - Matthew Vandermast, Nov 27 2009
a(n+1) = least common multiple of n-th row in A213999. - Reinhard Zumkeller, Jul 03 2012
For n > 2, (n-1) = Sum_{k=2..n} exp(a(n)*2*i*Pi/k). - Eric Desbiaux, Sep 13 2012
First column minus second column of A027446. - Eric Desbiaux, Mar 29 2013
For n > 0, a(n) is the smallest number k such that n is the n-th divisor of k. - Michel Lagneau, Apr 24 2014
Slowest growing integer > 0 in Z converging to 0 in Z^ when considered as profinite integer. - Herbert Eberle, May 01 2016
What is the largest number of consecutive terms that are all equal? I found 112 equal terms from a(370261) to a(370372). - Dmitry Kamenetsky, May 05 2019
Answer: there exist arbitrarily long sequences of consecutive terms with the same value; also, the maximal run of consecutive terms with different values is 5 from a(1) to a(5) (see link Roger B. Eggleton). - Bernard Schott, Aug 07 2019
Related to the inequality (54) in Ramanujan's paper about highly composite numbers A002182, also used in A199337: a(A329570(m))^2 is a (not minimal) bound above which all highly composite numbers are divisible by m, according to the right part of that inequality. - M. F. Hasler, Jan 04 2020
For n > 2, a(n) is of the form 2^e_1 * p_2^e_2 * ... * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) <= e_1. Therefore, 2^e * p_m^e_m is a primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 2, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m, is a Zumkeller number (see my proof at A002182 for details). - Ivan N. Ianakiev, May 10 2020
For n > 1, 2|(a(n)+2) ... n|(a(n)+n), so a(n)+2 .. a(n)+n are all composite and (part of) a prime gap of at least n.  (Compare n!+2 .. n!+n). - Stephen E. Witham, Oct 09 2021

			LCM of {1,2,3,4,5,6} = 60. The primes up to 6 are 2, 3 and 5. floor(log(6)/log(2)) = 2 so the exponent of 2 is 2.
floor(log(6)/log(3)) = 1 so the exponent of 3 is 1.
floor(log(6)/log(5)) = 1 so the exponent of 5 is 1. Therefore, a(6) = 2^2 * 3^1 * 5^1 = 60. - _David A. Corneth_, Jun 02 2017

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 365.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..2308 (first 501 terms from T. D. Noe)
R. Anderson and N. J. A. Sloane, Correspondence, 1975.
Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
Javier Cilleruelo, Juanjo Rué, Paulius Šarka, and Ana Zumalacárregui, The least common multiple of sets of positive integers, arXiv:1112.3013 [math.NT], 2011.
R. E. Crandall and C. Pomerance, Prime numbers: a computational perspective, MR2156291, p. 61.
Roger B. Eggleton, Least Common Multiple of {1,2,...,n}, Mathematics Magazine, 61(1) (1988), pp. 47-48, Problem 1252.
Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, arXiv:0906.2295 [math.NT], 2009.
Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, Amer. Math. Monthly 116(9) (2009), 836-839.
Steven Finch, Cilleruelo's LCM Constants, 2013. [Cached copy, with permission of the author]
V. L. Gavrikov, On property of least common multiple to be a D-magic number, arXiv:1806.09264 [math.NT], 2018.
S. Labbé and E. Pelantová, Palindromic sequences generated from marked morphisms, arXiv:1409.7510 [math.CO], 2014.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (6) (2002) 534-543. arXiv:math/0008177 [math.NT], 2000-2001.
Peter Luschny and S. Wehmeier, The lcm(1, 2, ..., n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, 99, pp 213-219 (2015).
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384 [math.CA], 2009.
M. Nair, On Chebychev-type inequalities for primes Amer. Math. Monthly 89(2) (1982), 126-129.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society ser. 2, vol. XIV, no. 1 (1915), pp 347-409. (A variant of a better quality with an additional footnote is available here.)
E. S. Selmer, On the number of prime divisors of a binomial coefficient, Math. Scand. 39 (1976), no. 2, 271-281 (1977).
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. AMS 131 (2003), 3335.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.
M. Tchebichef, Mémoire sur les nombres premiers, J. Math. Pures Appliquées 17 (1852), 366-390.
Helge von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1) (1901), 159-182.
Eric Weisstein's World of Mathematics, Least Common Multiple, Chebyshev Functions, Mangoldt Function.
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to lcm's

Row products of A133233.
Cf. A000142, A000793, A002110, A002182, A002201, A002944, A014963, A020500, A025527, A038610, A051173, A064446, A064859, A069513, A072938, A093880, A094348, A096179, A099996, A102910, A106037, A119682, A179661, A193181, A225558, A225630, A225632, A225640, A225642.
Cf. A025528 (number of prime factors of a(n) with multiplicity).
Cf. A275120 (lengths of runs of consecutive equal terms), A276781 (ordinal transform from term a(1)=1 onward).

a003418 = foldl lcm 1 . enumFromTo 2
-- Reinhard Zumkeller, Apr 04 2012, Apr 25 2011

[1] cat [Exponent(SymmetricGroup(n)) : n in [1..28]]; // Arkadiusz Wesolowski, Sep 10 2013

[Lcm([1..n]): n in [0..30]]; // Bruno Berselli, Feb 06 2015

A003418 := n-> lcm(seq(i,i=1..n));
HalfFarey := proc(n) local a,b,c,d,k,s; a := 0; b := 1; c := 1; d := n; s := NULL; do k := iquo(n + b, d); a, b, c, d := c, d, k*c - a, k*d - b; if 2*a > b then break fi; s := s,(a/b); od: [s] end: LCM := proc(n) local i; (1/2)*mul(2*sin(Pi*i),i=HalfFarey(n))^2 end: # Peter Luschny
# next Maple program:
a:= proc(n) option remember; `if`(n=0, 1, ilcm(n, a(n-1))) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Jun 10 2021

Table[LCM @@ Range[n], {n, 1, 40}] (* Stefan Steinerberger, Apr 01 2006 *)
FoldList[ LCM, 1, Range@ 28]
A003418[0] := 1; A003418[1] := 1; A003418[n_] := A003418[n] = LCM[n,A003418[n-1]]; (* Enrique Pérez Herrero, Jan 08 2011 *)
Table[Product[Prime[i]^Floor[Log[Prime[i], n]], {i, PrimePi[n]}], {n, 0, 28}] (* Wei Zhou, Jun 25 2011 *)
Table[Product[Cyclotomic[n, 1], {n, 2, m}], {m, 0, 28}] (* Fred Daniel Kline, May 22 2014 *)
a1[n_] := 1/12 (Pi^2+3(-1)^n (PolyGamma[1,1+n/2] - PolyGamma[1,(1+n)/2])) // Simplify
a[n_] := Denominator[Sqrt[a1[n]]];
Table[If[IntegerQ[a[n]], a[n], a[n]*(a[n])[[2]]], {n, 0, 28}] (* Gerry Martens, Apr 07 2018 [Corrected by Vaclav Kotesovec, Jul 16 2021] *)

a(n)=local(t); t=n>=0; forprime(p=2,n,t*=p^(log(n)\log(p))); t

a(n)=if(n<1,n==0,1/content(vector(n,k,1/k)))

a(n)=my(v=primes(primepi(n)),k=sqrtint(n),L=log(n+.5));prod(i=1,#v,if(v[i]>k,v[i],v[i]^(L\log(v[i])))) \\ Charles R Greathouse IV, Dec 21 2011

a(n)=lcm(vector(n,i,i)) \\ Bill Allombert, Apr 18 2012 [via Charles R Greathouse IV]

n=1; lim=100; i=1; j=1; until(n==lim, a=lcm(j,i+1); i++; j=a; n++; print(n" "a);); \\ Mike Winkler, Sep 07 2013

from functools import reduce
from operator import mul
from sympy import sieve
def integerlog(n,b): # find largest integer k>=0 such that b^k <= n
    kmin, kmax = 0,1
    while b**kmax <= n:
        kmax *= 2
    while True:
        kmid = (kmax+kmin)//2
        if b**kmid > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmin
def A003418(n):
    return reduce(mul,(p**integerlog(n,p) for p in sieve.primerange(1,n+1)),1) # Chai Wah Wu, Mar 13 2021

# generates initial segment of sequence
from math import gcd
from itertools import accumulate
def lcm(a, b): return a * b // gcd(a, b)
def aupton(nn): return [1] + list(accumulate(range(1, nn+1), lcm))
print(aupton(30)) # Michael S. Branicky, Jun 10 2021

[lcm(range(1,n)) for n in range(1, 30)] # Zerinvary Lajos, Jun 06 2009

(define (A003418 n) (let loop ((n n) (m 1)) (if (zero? n) m (loop (- n 1) (lcm m n))))) ;; Antti Karttunen, Jan 03 2018

The prime number theorem implies that lcm(1,2,...,n) = exp(n(1+o(1))) as n -> infinity. In other words, log(lcm(1,2,...,n))/n -> 1 as n -> infinity. - Jonathan Sondow, Jan 17 2005
a(n) = Product (p^(floor(log n/log p))), where p runs through primes not exceeding n (i.e., primes 2 through A007917(n)). - Lekraj Beedassy, Jul 27 2004
Greg Martin showed that a(n) = lcm(1,2,3,...,n) = Product_{i = Farey(n), 0 < i < 1} 2*Pi/Gamma(i)^2. This can be rewritten (for n > 1) as a(n) = (1/2)*(Product_{i = Farey(n), 0 < i <= 1/2} 2*sin(i*Pi))^2. - Peter Luschny, Aug 08 2009
Recursive formula useful for computations: a(0)=1; a(1)=1; a(n)=lcm(n,a(n-1)). - Enrique Pérez Herrero, Jan 08 2011
From Enrique Pérez Herrero, Jun 01 2011: (Start)
a(n)/a(n-1) = A014963(n).
if n is a prime power p^k then a(n)=a(p^k)=p*a(n-1), otherwise a(n)=a(n-1).
a(n) = Product_{k=2..n} (1 + (A007947(k)-1)*floor(1/A001221(k))), for n > 1. (End)
a(n) = A079542(n+1, 2) for n > 1.
a(n) = exp(Sum_{k=1..n} Sum_{d|k} moebius(d)*log(k/d)). - Peter Luschny, Sep 01 2012
a(n) = A025529(n) - A027457(n). - Eric Desbiaux, Mar 14 2013
a(n) = exp(Psi(n)) = 2 * Product_{k=2..A002088(n)} (1 - exp(2*Pi*i * A038566(k+1) / A038567(k))), where i is the imaginary unit, and Psi the second Chebyshev's function. - Eric Desbiaux, Aug 13 2014
a(n) = A064446(n)*A038610(n). - Anthony Browne, Jun 16 2016
a(n) = A000142(n) / A025527(n) = A000793(n) * A225558(n). - Antti Karttunen, Jun 02 2017
log(a(n)) = Sum_{k>=1} (A309229(n, k)/k - 1/k). - Mats Granvik, Aug 10 2019
From Petros Hadjicostas, Jul 24 2020: (Start)
Nair (1982) proved that 2^n <= a(n) <= 4^n for n >= 9. See also Farhi (2009). Nair also proved that
a(n) = lcm(m*binomial(n,m): 1 <= m <= n) and
a(n) = gcd(a(m)*binomial(n,m): n/2 <= m <= n). (End)
Sum_{n>=1} 1/a(n) = A064859. - Bernard Schott, Aug 24 2020

A094348 Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 360360, 498960, 554400, 665280, 720720

Offset: 1

Matthew Vandermast, Jun 18 2004, Oct 12 2008

nonn

Includes all highly composite numbers (A002182) and least common multiples of 1 through n (A003418). It would be interesting to know: 1) whether or not all deeply composite numbers (A095848, which includes all members of A003418) also belong to this sequence; 2) if 72 is the only member of this sequence not also belonging to A002182 or A095848.
465585120 is the first member of A095848 that is not a member of this sequence. The first members that belong to neither A002182 nor A095848 are 72, 30240, 64864800 and 1470268800. - David Wasserman, Jun 28 2007
This sequence is also A096179 with duplicates deleted and sorted. Let F(n) be the number of the row of A096179 which has the first occurrence of a(n) and M(n) = max{F(i),i <= n}. Then the following table indicates this connection.
  n |1,2,3,4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15, 16, 17,  18,  19,  20, 21
a(n)|1,2,4,6,12,24,36,48,60,72,120,180,240,360,420,720,840,1260,1680,2520,5040
F(n)|1,2,4,3, 4, 8,18,16, 5, 9,  8, 18, 16,  9,  7, 16, 15,  21,  16,   9, 16
M(n)|1,2,4,4, 4, 8,18,18,18,18, 18, 18, 18, 18, 18, 18, 18,  21,  21,  21, 21
- Peter Luschny, Dec 29 2010

			72 is a multiple of seven of the first nine positive integers (namely, 1, 2, 3, 4, 6, 8 and 9). It is the smallest positive integer for which this is true.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 844.
J. Britton, Perfect Number Analyzer
Wikipedia, Table of divisors.
Peter Luschny, More terms, OEIS wiki.

Cf. A096179, A003418, A002182, A002201, A072938, A106037, A002110.

\\ Computes the first 50 terms of A094348
A094348() = {local(n,i,R,A,len,count,change,high,lim);
lim = 7208000; R = vector(500); A = vector(50); A[1]=1;
A[2]=2; A[3]=4; A[4]=6; A[5]=12; count=5; high=0; n=12;
while(n < lim, d=divisors(n); len=length(d); change=0;
for(i=1,min(len,high),if(R[i]>d[i],R[i]=d[i];change=1));
if(len>high,for(i=high+1,len,R[i]=d[i]); high=len);
if(change, count++; A[count] = n); n += 12; );
write("A094348.txt", vector(count, i, A[i])); }
\\ Peter Luschny, Dec 29 2010

More terms from David Wasserman, Jun 28 2007

Title edited by Matthew Vandermast, Nov 20 2010

A181853 Triangle read by rows: T(n,k) = Sum_{c in C(n,k)} lcm(c) where C(n,k) is the set of all k-subsets of {1,2,...,n}.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 31, 34, 12, 1, 15, 81, 189, 182, 60, 1, 21, 141, 393, 494, 282, 60, 1, 28, 288, 1380, 3245, 3740, 2034, 420, 1, 36, 456, 2716, 8293, 13268, 11338, 4908, 840, 1, 45, 726, 5578, 22207, 47351, 57598, 40602, 15564, 2520

Offset: 0

Peter Luschny, Dec 06 2010

The C(n,k) are also called combinations of n with size k (see A181842).

Main diagonal gives: A003418. Lower diagonal gives: A094308. Column k=1 gives: A000217. - Alois P. Heinz, Jul 29 2013

			[0]   1
[1]   1    1
[2]   1    3     2
[3]   1    6    11     6
[4]   1   10    31    34    12
[5]   1   15    81   189   182    60
[6]   1   21   141   393   494   282   60

Alois P. Heinz, Rows n = 0..46, flattened

Row sums give A226037.

Cf. A065567, A096179, A181854.

with(combstruct):
a181853_row := proc(n) local k,L,l,R,comb;
R := NULL;
for k from 0 to n do
   L := 0;
   comb := iterstructs(Combination(n),size=k):
   while not finished(comb) do
      l := nextstruct(comb);
      L := L + ilcm(op(l));
   od;
   R := R,L;
od;
R end:
# second Maple program:
b:= proc(n, k) option remember; `if`(k=0, [1],
     [`if`(k add(c, c=b(n, k)):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jul 29 2013
# third Maple program:
b:= proc(n, m) option remember; expand(`if`(n=0, m,
      b(n-1, ilcm(m, n))*x+b(n-1, m)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..10);  # Alois P. Heinz, Sep 05 2023

t[, 0] = 1; t[n, k_] := Sum[LCM @@ c, {c, Subsets[Range[n], {k}]}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

# (After Alois P. Heinz)
@CachedFunction
def b(n, k):
    if k == 0: return [1]
    w = b(n-1, k) if kPeter Luschny, Jul 29 2013

A181854 Triangle read by rows: Partial row sums of A181853(n,k).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 7, 18, 24, 1, 11, 42, 76, 88, 1, 16, 97, 286, 468, 528, 1, 22, 163, 556, 1050, 1332, 1392, 1, 29, 317, 1697, 4942, 8682, 10716, 11136, 1, 37, 493, 3209, 11502, 24770, 36108, 41016, 41856

Offset: 0

Peter Luschny, Dec 06 2010

			[0]   1
[1]   1    2
[2]   1    4     6
[3]   1    7    18    24
[4]   1   11    42    76     88
[5]   1   16    97   286    468    528
[6]   1   22   163   556   1050   1332   1392

Alois P. Heinz, Rows n = 0..25, flattened

Cf. A096179, A181853.

with(combstruct):
a181854_row := proc(n) local k,L,l,R,comb;
R := NULL; L := 0;
for k from 0 to n do
   comb := iterstructs(Combination(n),size=k):
   while not finished(comb) do
      l := nextstruct(comb);
      L := L + ilcm(op(l));
   od;
   R := R,L;
od;
R end:

t[, 0] = 1; t[n, k_] := Sum[LCM @@ c, {c, Subsets[Range[n], {k}]}]; row[n_] := Table[t[n, k], {k, 0, n}] // Accumulate; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)

A179661 Triangle read by rows: T(n,k) is the largest least common multiple of any k-element subset of the first n positive integers.

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 4, 12, 12, 12, 5, 20, 60, 60, 60, 6, 30, 60, 60, 60, 60, 7, 42, 210, 420, 420, 420, 420, 8, 56, 280, 840, 840, 840, 840, 840, 9, 72, 504, 2520, 2520, 2520, 2520, 2520, 2520, 10, 90, 630, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 11, 110, 990

Offset: 1

Enrique Pérez Herrero, Jan 09 2011

Sequence differs from A093919; first divergences are at indices 31, 40, 48, 59.

Main diagonal is A003418.

			Triangle begins:
    [ 1 ],
    [ 2, 2 ],
    [ 3, 6, 6 ],
    [ 4, 12, 12, 12 ],
    [ 5, 20, 60, 60, 60 ],
    [ 6, 30, 60, 60, 60, 60 ].

Cf. A093919, A096179, A003418.

A179661:=func< n, k | Max([ LCM(s): s in Subsets({1..n}, k) ]) >; z:=12; [ A179661(n, k): k in [1..n], n in [1..z] ]; // Klaus Brockhaus, Jan 16 2011

A179661[n_,k_]:=Max[LCM@@@Subsets[Range[n],{k}]];
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A002024[n_]:=Floor[1/2+Sqrt[2*n]];
A179661[n_]:=A179661[A002024[n],A002260[n]]

T(n,k) = max{ lcm(x_1,...,x_k) ; 0 < x_1 < ... < x_k <= n }.

A182940 List of divisors of the sequence A094348 in their order of occurrence.

Original entry on oeis.org

1, 2, 4, 3, 6, 12, 8, 24, 9, 18, 36, 16, 48, 5, 10, 15, 20, 30, 60, 72, 40, 120, 45, 90, 180, 80, 240, 360, 7, 14, 21, 28, 35, 42, 70, 84, 105, 140, 210, 420, 144, 720, 56, 168, 280, 840

Offset: 1

Peter Luschny, Jan 04 2011

nonn

Read A094348 term-by-term, compute the divisors of each term, write down any divisors not seen before.

			1    [1]
2    [2]
4    [4]
6    [3, 6]
12   [12]
24   [8, 24]
36   [9, 18, 36]
48   [16, 48]
60   [5, 10, 15, 20, 30, 60]
72   [72]
120  [40, 120]
180  [45, 90, 180]
240  [80, 240]
360  [360]
420  [7, 14, 21, 28, 35, 42, 70, 84, 105, 140, 210, 420]
720  [144, 720]
840  [56, 168, 280, 840]
1260 [63, 126, 252, 315, 630, 1260]
1680 [112, 336, 560, 1680]
2520 [504, 2520]
5040 [1008, 5040]

Cf. A094348, A096179.

concat := (a,h)->[op(a),op(sort(convert(h,list)))];
DivisorsInOrder := proc(S) local A, H, T, s;
T := {}; A := [];
for s in S do
   H := numtheory[divisors](s) minus T:
   if H <> {} then
      A := concat(A, H);
      T := T union H
   fi
od;
A end:
A182940 := DivisorsInOrder(A094348);

cp's OEIS Frontend

A178938 Min{ n | A094348(n) element of {A096179(n,k) | 1 <= k <= n}}.

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A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.

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A094348 Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers.

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A181853 Triangle read by rows: T(n,k) = Sum_{c in C(n,k)} lcm(c) where C(n,k) is the set of all k-subsets of {1,2,...,n}.

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A181854 Triangle read by rows: Partial row sums of A181853(n,k).

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A179661 Triangle read by rows: T(n,k) is the largest least common multiple of any k-element subset of the first n positive integers.

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A182940 List of divisors of the sequence A094348 in their order of occurrence.

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