login
Maximum of the partition function on the set of all partitions of n.
1

%I #45 Jun 14 2022 07:06:13

%S 1,2,3,5,7,11,15,25,35,55,77,125,175,275,385,625,875,1375,1925,3125,

%T 4375,6875,9625,15625,21875,34375,48125,78125,109375,171875,240625,

%U 390625,546875,859375,1203125,1953125,2734375,4296875,6015625,9765625

%N Maximum of the partition function on the set of all partitions of n.

%C Sequence taken from the Bessenrodt-Ono paper, page 4, table 2, column of maxp(n), n = 1..14.

%C For the precise definition of maxp(n) see the theorem 1.1 in the mentioned paper, pages 1-2.

%C First differs from A000041 at a(8).

%H Christine Bessenrodt and Ken Ono, <a href="http://arxiv.org/abs/1403.3352">Maximal multiplicative properties of partitions</a>, arXiv:1403.3352 [math.CO], 2014.

%F a(n) = A000041(n), if 1 <= n <= 7.

%F a(n) = 5^(n/4), if n is congruent to 0 {mod 4} and n >= 8.

%F a(n) = 7*5^((n-5)/4), if n is congruent to 1 {mod 4} and n >= 8.

%F a(n) = 11*5^((n-6)/4), if n is congruent to 2 {mod 4} and n >= 8.

%F a(n) = 11*7*5^((n-11)/4), if n is congruent to 3 {mod 4} and n >= 8.

%t a[n_] := If[n <= 7, PartitionsP[n], Switch[Mod[n, 4], 0, 5^(n/4), 1, 7*5^((n-5)/4), 2, 11*5^((n-6)/4), 3, 11*7*5^((n-11)/4)]];

%t Array[a, 40] (* _Jean-François Alcover_, Sep 23 2018 *)

%o (PARI) a(n) = {if (n <= 7, numbpart(n), my(nmf = n % 4); if (nmf == 0, 5^(n/4), if (nmf == 1, 7*5^((n-5)/4), if (nmf == 2, 11*5^((n-6)/4), 11*7*5^((n-11)/4)););););} \\ _Michel Marcus_, Mar 16 2014

%Y Cf. A000041, A239314.

%K nonn

%O 1,2

%A _Omar E. Pol_, Mar 15 2014

%E a(22) corrected by _Michel Marcus_, Jun 14 2022