A002809 -id:A002809 - cp's OEIS frontend

A002497 Numbers N in A002809 such that there is rho > 0 such that for all A > 0, A008475(A)-A008475(N) >= rho*log(A/N).

3, 12, 60, 420, 4620, 60060, 180180, 360360, 6126120, 116396280, 2677114440, 77636318760, 2406725881560, 89048857617720, 3651003162326520, 156993135980040360, 313986271960080720, 14757354782123793840, 14757354782123793840, 782139803452561073520, 46146248403701103337680

Offset: 1

N. J. A. Sloane

The numbers contain the starred entries on pp. 187-190 of Nicolas. It is a subsequence of A002809 by selecting only elements of a set/property "G" (page 150). G contains all N such that a real, strictly positive rho exists such that for all strictly positive integers A we have l(A)-l(N) >= rho*log(A/N). The function l()=A008475() is defined on page 139. - R. J. Mathar, Mar 23 2012

These numbers were named superior l-composite numbers (nombres l-composes superieurs, the function l(n) is A002809) by Massias, in analogy to Ramanujan's superior highly composite numbers (A002201). Deléglise and Nicolas named these numbers l-superchampion numbers. They are used by Deléglise et al. in calculating values of Landau's function g(n) (A000793). - Amiram Eldar, Aug 23 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marc Deléglise and Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, arXiv preprint, arXiv:1907.07664 [math.NT] (2019).
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, Journal de théorie des nombres de Bordeaux, Vol. 20. No. 3 (2008), pp. 625-671.
Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, Annales de la Faculté des Sciences de Toulouse: Mathématiques, Vol. 6, No. 3-4 (1984), pp. 269-281.
Jean-Louis Nicolas, Ordre maximum d'un élément du groupe S(n) des permutations et "highly composite numbers", Bull. Soc. Math. Française 97 (1969), 129-191.

Cf. A000793, A002201, A002498, A002809, A008475.

Edited by M. F. Hasler, Mar 29 2015

a(16)-a(21) from the paper by Massias added by Amiram Eldar, Aug 23 2019

A159685 Maximal product of distinct primes whose sum is <= n.

Original entry on oeis.org

1, 2, 3, 3, 6, 6, 10, 15, 15, 30, 30, 42, 42, 70, 105, 105, 210, 210, 210, 210, 330, 330, 462, 462, 770, 1155, 1155, 2310, 2310, 2730, 2730, 2730, 2730, 4290, 4290, 6006, 6006, 10010, 15015, 15015, 30030, 30030, 30030, 30030, 39270, 39270, 46410, 46410

Offset: 1

Wouter Meeussen, Apr 19 2009, May 02 2009

nonn

Equivalently, largest value of the LCM of the partitions of n into primes.
Equivalently, maximal number of times a permutation of length n, with prime cycle lengths, can operate on itself before returning to the initial permutation.
If the requirement that primes are distinct is dropped, this becomes A000792. - Charles R Greathouse IV, Jul 10 2012

			A permutation of length 10 can have prime cycle lengths of 2+3+5; so when repeatedly applied to itself, can produce at most 2*3*5 different permutations.
The products of distinct primes whose sum is <= 10 are 1 (the empty product), 2, 3, 5, 7, 2*3=6, 2*5=10, 2*7=14, 3*5=15, 3*7=21, and 2*3*5=30. The maximum is 30, so a(10) = 30. - _Jonathan Sondow_, Jul 06 2012

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. Deléglise and J.-L. Nicolas, Maximal product of primes whose sum is bounded, arXiv 1207.0603 [math.NT] (2012).
Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
Marc Deléglise, Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, Univ. Lyon (France, 2019).

Cf. A077011, A000793, A034891.

with(numtheory):
b:= proc(n,i) option remember; local p; p:= ithprime(max(i,1));
      `if`(n=0, 1, `if`(i<1, 0,
       max(b(n, i-1), `if`(p>n, 0, b(n-p, i-1)*p))))
    end:
a:= proc(n) option remember;
     `if`(n=0, 1, max(b(n, pi(n)), a(n-1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 04 2012

temp=Series[Times @@ (1/(1-q[ # ] x^#)& /@ Prepend[Prime /@ Range[24],1]),{x,0,Prime[24]}]; Table[Max[List @@ Expand[Coefficient[temp,x^n]]/. q[a_]^_ ->q[a] /.q->Identity],{n,64}]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{p = Prime[Max[i, 1]]}, If[n == 0, 1, If[i < 1, 0, Max[b[n, i-1], If[p > n, 0, b[n-p, i-1]*p]]]]]; a[n_] := a[n] = If[n == 0, 1, Max[b[n, PrimePi[n]], a[n-1]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 05 2013, translated from Alois P. Heinz's Maple program *)

a(n) <= A002809(n) and A008475(a(n)) <= n (see (1.2) and (1.4) in Deléglise-Nicolas 2012). - Jonathan Sondow, Jul 04 2012.

A006644 Indices of records in Landau's function A000793.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 16, 17, 19, 23, 25, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 47, 49, 53, 57, 58, 59, 60, 62, 64, 66, 68, 70, 72, 76, 77, 78, 79, 83, 85, 89, 93, 95, 97, 101, 102, 106, 108, 112, 118, 120, 126, 128, 130, 131, 132

Offset: 1

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3318 from Alois P. Heinz)
J.-L. Nicolas, Ordre maximum d'un élément du groupe de permutations et highly composite numbers, Bull. Math. Soc. France, 97 (1969), 129-191.

Cf. A000793, A002809.

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}]]]]; order[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Join[{0}, Position[Differences[Array[order, 133, 0]], ?Positive] // Flatten ] (* _Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

A211168 Exponent of alternating group An.

Original entry on oeis.org

1, 1, 3, 6, 30, 60, 420, 420, 1260, 2520, 27720, 27720, 360360, 360360, 360360, 360360, 6126120, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 1

Alexander Gruber, Jan 31 2013

a(n) is the smallest natural number m such that g^m = 1 for any g in An.

If m <= n, a m-cycle occurs in some permutation in An if and only if m is odd or m <= n - 2. The exponent is the LCM of the m's satisfying these conditions, leading to the formula below.

			For n = 7, lcm{1,...,5,7} = 420.

Alexander Gruber, Table of n, a(n) for n = 1..2308

Even entries given by the sequence A076100, or the odd entries in the sequence A003418.

The records of this sequence are a subsequence of A002809 and A126098.

for n in [1..40] do
Exponent(AlternatingGroup(n));
end for;

for n in [1..40] do
if n mod 2 eq 0 then
L := [1..n-1];
else
L := Append([1..n-2],n);
end if;
LCM(L);
end for;

Table[If[Mod[n, 2] == 0, LCM @@ Range[n - 1],
  LCM @@ Join[Range[n - 2], {n}]], {n, 1, 100}] (* or *)
a[1] = 1; a[2] = 1; a[3] = 3; a[n_] := a[n] =
  If[Mod[n, 2] == 0, LCM[a[n - 1], n - 2], LCM[a[n - 2], n - 3, n]]; Table[a[n], {n, 1, 40}]

a(n)=lcm(if(n%2,concat([2..n-2],n),[2..n-1])) \\ Charles R Greathouse IV, Mar 02 2014

Explicit:
a(n) = lcm{1, ..., n-1} if n is even.
     = lcm{1, ..., n-2, n} if n is odd.
Recursive:
Let a(1) = a(2) = 1 and a(3) = 3.  Then
a(n) = lcm{a(n-1), n-2} if n is even.
     = lcm{a(n-2), n-3, n} if n is odd.
a(n) = A003418(n)/(1 + [n in A228693]) for n > 1. - Charlie Neder, Apr 25 2019

A214096 Smallest m such that prime(i) + prime(i-1) < prime(2*i-n) for all i>=m.

Original entry on oeis.org

3, 4, 7, 8, 18, 19, 27, 28, 36, 39, 50, 50, 53, 70, 71, 72, 77, 85, 105, 105, 106, 108, 110, 111, 114, 143, 144, 144, 149, 149, 153, 161, 165, 172, 173, 173, 226, 228, 228, 229, 231, 232, 236, 237, 238, 245, 245, 246, 248, 300, 300, 301, 302, 303, 315, 315

Offset: 1

Jonathan Vos Post, Jul 04 2012

nonn

Formula given in Deléglise and Nicolas, Lemma 2.4, p.6. A002809 and A159685 are given explicitly on p.2. Additional values given: a(3675) = 33127.

Jean-François Alcover, Table of n, a(n) for n = 1..3675
Marc Deléglise, Jean-Louis Nicolas, Maximal product of primes whose sum is bounded, arXiv:1207.0603v1 [math.NT], June 3, 2012.

Cf. A000040, A002809, A159685.

a[1] = 3;
a[n_] := a[n] = Module[{}, For[m = a[n-1], True, m++, If[AllTrue[Range[m, 2 m], Prime[#] + Prime[# - 1] < Prime[2# - n]&], Return[m]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 27 2018 *)

a(n) is minimal such that prime(i) + prime(i-1) < prime(2*i-n) for i >= a(n).

More terms from Alois P. Heinz, Jul 07 2012

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A002497 Numbers N in A002809 such that there is rho > 0 such that for all A > 0, A008475(A)-A008475(N) >= rho*log(A/N).

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A159685 Maximal product of distinct primes whose sum is <= n.

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A006644 Indices of records in Landau's function A000793.

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A211168 Exponent of alternating group An.

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A214096 Smallest m such that prime(i) + prime(i-1) < prime(2*i-n) for all i>=m.

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