A332644 Largest of the least integers of prime signatures over all partitions of n into distinct parts.

1, 2, 4, 12, 24, 72, 360, 720, 2160, 10800, 75600, 151200, 453600, 2268000, 15876000, 174636000, 349272000, 1047816000, 5239080000, 36673560000, 403409160000, 5244319080000, 10488638160000, 31465914480000, 157329572400000, 1101307006800000, 12114377074800000

Offset: 0

Author

Alois P. Heinz, Feb 18 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..712
• Eric Weisstein's World of Mathematics, Prime Signature
• Wikipedia, Partition (number theory)
• Wikipedia, Prime signature
• Index entries for sequences related to prime signature

Crossrefs

Subsequence of A025487, A055932, A087980.

Cf. A000009, A000040, A000217, A002110, A002260, A003056, A001221, A001222, A007814, A037126, A046523, A123578, A328524.

Programs

Maple

b:= proc(n, i, j) option remember;
      `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
      ithprime(n-(t-> t*(t+1)/2)(floor((sqrt(8*n-7)-1)/2))))
    end:
seq(a(n), n=0..30);

Mathematica

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, 0, If[n == 0, 1, Max[b[n, i - 1, j], Prime[j]^i b[n - i, Min[n - i, i - 1], j + 1]]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 30] (* Jean-François Alcover, May 07 2020, after 1st Maple program *)

Formula

a(n) = A328524(n,A000009(n)).

A001221(a(n)) = A003056(n).

A001222(a(n)) = n.

A046523(a(n)) = a(n).

a(n)/a(n-1) = A037126(n) = A000040(n-A000217(A003056(n))) for n > 0.

a(n) in { A025487 }.

a(n) in { A055932 }.

a(n) in { A087980 }.

A007814(a(n)) = A123578(n).