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.

0, 1, 6, 23, 64, 129, 222, 345, 502, 695, 924, 1193, 1502, 1853, 2246, 2687, 3172, 3705, 4286, 4917, 5600, 6333, 7118, 7957, 8848, 9797, 10800, 11861, 12978, 14153, 15386, 16681, 18034, 19447, 20922, 22459, 24060, 25723, 27448, 29239, 31094, 33015

Offset: 0

Author

Moshe Shmuel Newman, Jun 11 2006

Keywords

• nonn

Comments

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.

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

Links

• Jean-François Alcover, Table of n, a(n) for n = 0..60
• Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
• I. J. Good, Problem 6137, American Mathematical Monthly, 1978, pages 830-831.
• Hansraj Gupta, Finite Differences of the Partition Function, Math. Comp. 32 (1978), 1241-1243.
• A. M. Odlyzko, Differences of the partition function, Acta Arithmetica 49.3 (1988): 237-254.
• Eric Weisstein's World of Mathematics, Forward Difference.

Crossrefs

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

Programs

Maple

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

Mathematica

a[n_] := a[n] = Module[{f}, f[i_] = DifferenceDelta[PartitionsP[i], {i, n}]; For[j = 2, True, j++, If[f[j] > 0 && f[j+1] > 0, Return[j]]]];
a[0] = 0; a[1] = 1;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 04 2020 *)

Formula

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

Extensions

a(11)-a(41) from Alois P. Heinz, Jul 20 2009