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

%I A332644 #48 Feb 16 2025 08:33:59
%S A332644 1,2,4,12,24,72,360,720,2160,10800,75600,151200,453600,2268000,
%T A332644 15876000,174636000,349272000,1047816000,5239080000,36673560000,
%U A332644 403409160000,5244319080000,10488638160000,31465914480000,157329572400000,1101307006800000,12114377074800000
%N A332644 Largest of the least integers of prime signatures over all partitions of n into distinct parts.
%H A332644 Alois P. Heinz, <a href="/A332644/b332644.txt">Table of n, a(n) for n = 0..712</a>
%H A332644 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>
%H A332644 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A332644 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_signature">Prime signature</a>
%H A332644 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%F A332644 a(n) = A328524(n,A000009(n)).
%F A332644 A001221(a(n)) = A003056(n).
%F A332644 A001222(a(n)) = n.
%F A332644 A046523(a(n)) = a(n).
%F A332644 a(n)/a(n-1) = A037126(n) = A000040(n-A000217(A003056(n))) for n > 0.
%F A332644 a(n) in { A025487 }.
%F A332644 a(n) in { A055932 }.
%F A332644 a(n) in { A087980 }.
%F A332644 A007814(a(n)) = A123578(n).
%p A332644 b:= proc(n, i, j) option remember;
%p A332644       `if`(i*(i+1)/2<n, 0, `if`(n=0, 1, max(b(n, i-1, j),
%p A332644        ithprime(j)^i*b(n-i, min(n-i, i-1), j+1))))
%p A332644     end:
%p A332644 a:= n-> b(n$2, 1):
%p A332644 seq(a(n), n=0..30);
%p A332644 # second Maple program:
%p A332644 a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
%p A332644       ithprime(n-(t-> t*(t+1)/2)(floor((sqrt(8*n-7)-1)/2))))
%p A332644     end:
%p A332644 seq(a(n), n=0..30);
%t A332644 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]]]];
%t A332644 a[n_] := b[n, n, 1];
%t A332644 a /@ Range[0, 30] (* _Jean-François Alcover_, May 07 2020, after 1st Maple program *)
%Y A332644 Subsequence of A025487, A055932, A087980.
%Y A332644 Cf. A000009, A000040, A000217, A002110, A002260, A003056, A001221, A001222, A007814, A037126, A046523, A123578, A328524.
%K A332644 nonn
%O A332644 0,2
%A A332644 _Alois P. Heinz_, Feb 18 2020