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

Author

William J. Keith, Mar 18 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

Crossrefs

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

Programs

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

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)).