login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Largest of the least integers of prime signatures over all partitions of n into distinct parts.
2

%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