A080744 -id:A080744 - cp's OEIS frontend

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

1, 0, 2, 3, 4, 6, 0, 12, 15, 20, 30, 28, 60, 40, 84, 105, 140, 210, 180, 420, 280, 330, 360, 840, 504, 1260, 1155, 1540, 2310, 2520, 4620, 3080, 5460, 3960, 9240, 5544, 13860, 6552, 16380, 15015, 27720, 30030, 32760, 60060, 40040, 45045, 51480, 120120

Offset: 0

Vladeta Jovovic

nonn

			a(11) = 28 because max{11, 2*3^2, 2^3*3, 2^2*7} = 28.

Robert Gerbicz, Table of n, a(n) for n = 0..1000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Largest element of n-th row of A080743.
A000793(n)=max{A000793(n-1), a(n)}, A000793(0)=1.
Cf. A008475, A051613, A080743, A080744, A051704.

b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0, 1, `if`(i<1 or n<0, 0, max(b(n, i-1),
      seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))) ))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 16 2013

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 *)

{N=1000;v=vector(N,i,0);forprime(p=2,N,q=1;forstep(i=N,1,-1,
q=1;while(q*p<=i,q*=p;if(i==q,M=q,if(v[i-q],M=q*v[i-q],M=0));
v[i]=max(v[i],M))));print(0" "1);for(i=1,N,print(i" "v[i]))} \\ Robert Gerbicz, Jul 31 2012

Corrected and extended by Robert Gerbicz, Jul 31 2012

A080743 Array read by rows in which n-th row lists orders of elements of Symm(n) that are not orders of elements of Symm(n-1) (6th row is empty, written as 0 by convention).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 0, 7, 10, 12, 8, 15, 9, 14, 20, 21, 30, 11, 18, 24, 28, 35, 42, 60, 13, 22, 36, 40, 33, 45, 70, 84, 26, 44, 56, 105, 16, 39, 55, 63, 66, 90, 120, 140, 17, 52, 72, 210, 65, 77, 78, 110, 126, 132, 168, 180, 19, 34, 48, 88, 165, 420, 51, 91, 99, 130, 154, 156, 220

Offset: 1

N. J. A. Sloane, Mar 08 2003

A051613 gives number of elements in n-th row.

			1;
2;
3;
4;
5, 6;
0;
7, 10, 12;
8, 15;
...

Alois P. Heinz, Rows n = 1..100, flattened
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A008475, A051613, A080744, A051703.

b:= proc(n) option remember; `if`(n<3, {n},
      {n, seq(map(x-> ilcm(x, i), b(n-i))[], i=2..n-1)}
       minus {seq(b(i)[], i=1..n-1)})
    end:
T:= proc(n) local l; l:= [b(n, n)[]];
      `if`(nops(l)=0, 0, sort(l)[])
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 15 2013

b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0, 1, If[i<1 || n<0, 0, Max[Join[{b[n, i-1]}, Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n]}]]]]] ]; T[1] = {1}; T[6] = {0}; T[n_] := Reap[For[m = n, m <= b[n, PrimePi[n]], m++,  If[n == Total[Power @@@ FactorInteger[m]], Sow[m]]]][[2, 1]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

n-th row = set of m such that A008475(m) = n, or 0 if no such m exists.

More terms from Vladeta Jovovic, Mar 12 2003

cp's OEIS Frontend

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

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