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A119712 a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards. 2


%S 0,1,6,23,64,129,222,345,502,695,924,1193,1502,1853,2246,2687,3172,

%T 3705,4286,4917,5600,6333,7118,7957,8848,9797,10800,11861,12978,14153,

%U 15386,16681,18034,19447,20922,22459,24060,25723,27448,29239,31094,33015

%N a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards.

%C The first entry is considered to be indexed by zero. For example, the third difference A072380 starts with -1,1 and continues alternating in sign till the 24th entry, from which point it is positive.

%C Using a different (backward) definition of the difference operator, this sequence has also been given as 1,8,26,68,134,228,352,... A155861.

%D I. J. Good, Problem 6137, American Mathematical Monthly 1978 pages 830-831.

%D Hansraj Gupta, Finite Differences of the Partition Function, pp. 1241-1243.

%D A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254.

%H Almkvist, Gert, "<a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6126.pdf">On the differences of the partition function</a>", Acta Arith., 61.2 (1992), 173-181.

%H Weisstein, Eric W. "<a href="http://mathworld.wolfram.com/ForwardDifference.html">Forward Difference</a>".

%F Odlyzko gives an asymptotic formula a(n)~(6/(Pi)^2) * (n log n)^2

%p with(combinat): DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end: a:= proc(n) option remember; local f, k; if n=0 then 0 else f:= (DD@@n)(numbpart); for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k fi end: seq(a(n), n=0..20); # _Alois P. Heinz_, Jul 20 2009

%Y Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A155861.

%K nonn

%O 0,3

%A _Moshe Shmuel Newman_, Jun 11 2006

%E a(11)-a(41) from _Alois P. Heinz_, Jul 20 2009

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Last modified December 19 06:03 EST 2018. Contains 318245 sequences. (Running on oeis4.)