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 A159685 Maximal product of distinct primes whose sum is <= n. 3

%I

%S 1,2,3,3,6,6,10,15,15,30,30,42,42,70,105,105,210,210,210,210,330,330,

%T 462,462,770,1155,1155,2310,2310,2730,2730,2730,2730,4290,4290,6006,

%U 6006,10010,15015,15015,30030,30030,30030,30030,39270,39270,46410,46410

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

%C Equivalently, largest value of the LCM of the partitions of n into primes.

%C Equivalently, maximal number of times a permutation of length n, with prime cycle lengths, can operate on itself before returning to the initial permutation.

%C If the requirement that primes are distinct is dropped, this becomes A000792. - _Charles R Greathouse IV_, Jul 10 2012

%H Alois P. Heinz, <a href="/A159685/b159685.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Deléglise and J.-L. Nicolas, <a href="http://arxiv.org/abs/1207.0603">Maximal product of primes whose sum is bounded</a>, arXiv 1207.0603 [math.NT] (2012).

%H Marc Deléglise and Jean-Louis Nicolas, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Deleglise/deleglise3.html">On the Largest Product of Primes with Bounded Sum</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.

%H Marc Deléglise, Jean-Louis Nicolas, <a href="https://hal.archives-ouvertes.fr/hal-02177338/">The Landau function and the Riemann hypothesis</a>, Univ. Lyon (France, 2019).

%F a(n) <= A002809(n) and A008475(a(n)) <= n (see (1.2) and (1.4) in Deléglise-Nicolas 2012). - _Jonathan Sondow_, Jul 04 2012.

%e 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.

%e 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

%p with(numtheory):

%p b:= proc(n,i) option remember; local p; p:= ithprime(max(i,1));

%p `if`(n=0, 1, `if`(i<1, 0,

%p max(b(n, i-1), `if`(p>n, 0, b(n-p, i-1)*p))))

%p end:

%p a:= proc(n) option remember;

%p `if`(n=0, 1, max(b(n, pi(n)), a(n-1)))

%p end:

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jun 04 2012

%t temp=Series[Times @@ (1/(1-q[ # ] x^#)& /@ Prepend[Prime /@ Range,1]),{x,0,Prime}]; Table[Max[List @@ Expand[Coefficient[temp,x^n]]/. q[a_]^_ ->q[a] /.q->Identity],{n,64}]

%t (* Second program: *)

%t 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 *)

%Y Cf. A077011, A000793, A034891.

%K nonn

%O 1,2

%A _Wouter Meeussen_, Apr 19 2009, May 02 2009

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Last modified September 21 14:54 EDT 2020. Contains 337272 sequences. (Running on oeis4.)