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%I #47 May 07 2020 10:02:57
%S 1,2,4,12,24,72,360,720,2160,10800,75600,151200,453600,2268000,
%T 15876000,174636000,349272000,1047816000,5239080000,36673560000,
%U 403409160000,5244319080000,10488638160000,31465914480000,157329572400000,1101307006800000,12114377074800000
%N Largest of the least integers of prime signatures over all partitions of n into distinct parts.
%H Alois P. Heinz, <a href="/A332644/b332644.txt">Table of n, a(n) for n = 0..712</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_signature">Prime signature</a>
%H <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%F a(n) = A328524(n,A000009(n)).
%F A001221(a(n)) = A003056(n).
%F A001222(a(n)) = n.
%F A046523(a(n)) = a(n).
%F a(n)/a(n-1) = A037126(n) = A000040(n-A000217(A003056(n))) for n > 0.
%F a(n) in { A025487 }.
%F a(n) in { A055932 }.
%F a(n) in { A087980 }.
%F A007814(a(n)) = A123578(n).
%p b:= proc(n, i, j) option remember;
%p `if`(i*(i+1)/2<n, 0, `if`(n=0, 1, max(b(n, i-1, j),
%p ithprime(j)^i*b(n-i, min(n-i, i-1), j+1))))
%p end:
%p a:= n-> b(n$2, 1):
%p seq(a(n), n=0..30);
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
%p ithprime(n-(t-> t*(t+1)/2)(floor((sqrt(8*n-7)-1)/2))))
%p end:
%p seq(a(n), n=0..30);
%t 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 a[n_] := b[n, n, 1];
%t a /@ Range[0, 30] (* _Jean-François Alcover_, May 07 2020, after 1st Maple program *)
%Y Subsequence of A025487, A055932, A087980.
%Y Cf. A000009, A000040, A000217, A002110, A002260, A003056, A001221, A001222, A007814, A037126, A046523, A123578, A328524.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Feb 18 2020