A239445 Values n at which ratios of successive partition numbers approach 1 closer than the reciprocal of a whole number.

2, 3, 11, 25, 39, 57, 78, 102, 130, 161, 195, 232, 273, 317, 365, 415, 469, 526, 587, 651, 718, 788, 862, 939, 1019, 1103, 1189, 1280, 1373, 1470, 1570, 1673, 1779, 1889, 2002, 2119, 2239, 2362, 2488, 2618, 2750, 2887, 3026, 3169, 3315, 3464, 3617, 3773, 3932, 4094, 4260, 4429, 4602, 4777, 4956

Offset

1,1

Comments

The ratios of successive partition numbers p(n) / p(n-1) approach 1 monotonically, for n>1. a(k) gives the n for which p(n)/p(n+1) first equals or is less than 1+1/k.

Links

Table of n, a(n) for n=1..55.

Formula

Empirical quadratic fit to first 78 terms: ak^2 + bk + c, a ~ 1.64466, b ~ -0.3287, c ~ -0.66.

Leading term appears to approach 1.644... k^2, where the constant is zeta(2), Pi^2/6. This can probably be rigorously derived from the asymptotic expansion of the partition function, p(n) ~ 1/(4 n sqrt(3)) exp( Pi sqrt(2n/3)).

Example

p(2)=2 and p(1)=1, so a(1) = 2, since p(2)/p(1) = 1+1/1.
p(3)=3 and p(2)=2, so a(2)=3, since p(3)/p(2) = 1+1/2.
p(11)=56 and p(10) = 42, so a(3) = 11, since p(11)/p(10) = 1+1/3.

Mathematica

AddDenom = 2;
Breaks = {};
For[n = 2, n < 10000, n++,
If[PartitionsP[n]/PartitionsP[n - 1] <= (1 + (1/AddDenom)),
  AppendTo[Breaks, n]; ADH = AddDenom + 1; AddDenom = ADH]
]
Breaks

Crossrefs

Cf. A000041 (Partition numbers), A013661 (Pi^2 / 6).

Sequence in context: A136402 A137811 A041955 * A157161 A041811 A229066

Adjacent sequences: A239442 A239443 A239444 * A239446 A239447 A239448

Keyword

nonn,easy

Author

William J. Keith, Mar 18 2014

Status

approved