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%I #8 Mar 22 2014 14:14:11
%S 2,3,11,25,39,57,78,102,130,161,195,232,273,317,365,415,469,526,587,
%T 651,718,788,862,939,1019,1103,1189,1280,1373,1470,1570,1673,1779,
%U 1889,2002,2119,2239,2362,2488,2618,2750,2887,3026,3169,3315,3464,3617,3773,3932,4094,4260,4429,4602,4777,4956
%N Values n at which ratios of successive partition numbers approach 1 closer than the reciprocal of a whole number.
%C 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.
%F Empirical quadratic fit to first 78 terms: ak^2 + bk + c, a ~ 1.64466, b ~ -0.3287, c ~ -0.66.
%F 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)).
%e p(2)=2 and p(1)=1, so a(1) = 2, since p(2)/p(1) = 1+1/1.
%e p(3)=3 and p(2)=2, so a(2)=3, since p(3)/p(2) = 1+1/2.
%e p(11)=56 and p(10) = 42, so a(3) = 11, since p(11)/p(10) = 1+1/3.
%t AddDenom = 2;
%t Breaks = {};
%t For[n = 2, n < 10000, n++,
%t If[PartitionsP[n]/PartitionsP[n - 1] <= (1 + (1/AddDenom)),
%t AppendTo[Breaks, n]; ADH = AddDenom + 1; AddDenom = ADH]
%t ]
%t Breaks
%Y Cf. A000041 (Partition numbers), A013661 (Pi^2 / 6).
%K nonn,easy
%O 1,1
%A _William J. Keith_, Mar 18 2014
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