cp's OEIS Frontend

A051703 Maximal value of products of partitions of n into powers of distinct primes (1 not considered a power).

%I A051703 #24 Oct 02 2017 10:54:56
%S A051703 1,0,2,3,4,6,0,12,15,20,30,28,60,40,84,105,140,210,180,420,280,330,
%T A051703 360,840,504,1260,1155,1540,2310,2520,4620,3080,5460,3960,9240,5544,
%U A051703 13860,6552,16380,15015,27720,30030,32760,60060,40040,45045,51480,120120
%N A051703 Maximal value of products of partitions of n into powers of distinct primes (1 not considered a power).
%H A051703 Robert Gerbicz, <a href="/A051703/b051703.txt">Table of n, a(n) for n = 0..1000</a>
%H A051703 J. Bamberg, G. Cairns and D. Kilminster, <a href="http://www.jstor.org/stable/3647934">The crystallographic restriction, permutations and Goldbach's conjecture</a>, Amer. Math. Monthly, 110 (March 2003), 202-209.
%e A051703 a(11) = 28 because max{11, 2*3^2, 2^3*3, 2^2*7} = 28.
%p A051703 b:= proc(n, i) option remember; local p;
%p A051703       p:= `if`(i<1, 1, ithprime(i));
%p A051703       `if`(n=0, 1, `if`(i<1 or n<0, 0, max(b(n, i-1),
%p A051703       seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))) ))
%p A051703     end:
%p A051703 a:= n-> b(n, numtheory[pi](n)):
%p A051703 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 16 2013
%t A051703 nmax = 48; Do[a[n]=0, {n, 1, nmax}]; km = PrimePi[nmax]; For[k=1, k <= km, k++, q = 1; p = Prime[k]; For[i=nmax, i >= 1, i--, q=1; While[q*p <= i, q *= p; If[i == q, m = q, If[a[i - q] != 0, m = q*a[i - q], m = 0]]; a[i] = Max[a[i], m]]]]; a[0] = 1; Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Aug 02 2012, translated from _Robert Gerbicz_'s Pari program *)
%o A051703 (PARI) {N=1000;v=vector(N,i,0);forprime(p=2,N,q=1;forstep(i=N,1,-1,
%o A051703 q=1;while(q*p<=i,q*=p;if(i==q,M=q,if(v[i-q],M=q*v[i-q],M=0));
%o A051703 v[i]=max(v[i],M))));print(0" "1);for(i=1,N,print(i" "v[i]))} \\ _Robert Gerbicz_, Jul 31 2012
%Y A051703 Largest element of n-th row of A080743.
%Y A051703 A000793(n)=max{A000793(n-1), a(n)}, A000793(0)=1.
%Y A051703 Cf. A008475, A051613, A080743, A080744, A051704.
%K A051703 nonn
%O A051703 0,3
%A A051703 _Vladeta Jovovic_
%E A051703 Corrected and extended by _Robert Gerbicz_, Jul 31 2012