A239054 Maximum of the partition function on the set of all partitions of n.

1, 2, 3, 5, 7, 11, 15, 25, 35, 55, 77, 125, 175, 275, 385, 625, 875, 1375, 1925, 3125, 4375, 6875, 9625, 15625, 21875, 34375, 48125, 78125, 109375, 171875, 240625, 390625, 546875, 859375, 1203125, 1953125, 2734375, 4296875, 6015625, 9765625

Offset

1,2

Comments

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

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

First differs from A000041 at a(8).

Links

Table of n, a(n) for n=1..40.

Christine Bessenrodt and Ken Ono, Maximal multiplicative properties of partitions, arXiv:1403.3352 [math.CO], 2014.

Formula

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

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

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

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

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

Mathematica

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)]];
Array[a, 40] (* Jean-François Alcover, Sep 23 2018 *)

Prog

(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

Crossrefs

Cf. A000041, A239314.

Sequence in context: A177485 A218023 A165801 * A241725 A022480 A308927

Adjacent sequences: A239051 A239052 A239053 * A239055 A239056 A239057

Keyword

nonn

Author

Omar E. Pol, Mar 15 2014

Extensions

a(22) corrected by Michel Marcus, Jun 14 2022

Status

approved