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A239445 Values n at which ratios of successive partition numbers approach 1 closer than the reciprocal of a whole number. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 22 08:25 EDT 2019. Contains 322329 sequences. (Running on oeis4.)