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A274383 a(n) is the least m such that A008284(m,n+1) > A008284(m,n). 0
4, 7, 10, 15, 18, 23, 29, 35, 40, 47, 54, 60, 68, 75, 83, 90, 99, 107, 116, 125, 134, 143, 152, 162, 172, 182, 193, 203, 214, 225, 236, 248, 259, 271, 283, 295, 307, 320, 332, 345, 358, 372, 385, 398, 412, 426, 440, 454, 469, 483, 498, 513, 528, 543, 559, 574, 590, 606, 622, 638, 654, 671, 688, 704 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A008284(m,n) is the number of partitions of the integer m into n parts; p(m,n) in the following. It is numerically and intuitively clear that for any fixed n, for sufficiently large m, p(m,n+1) > p(m,n).  Moreover, from examining the table of p(m,n) for small values of n, it appears that for any fixed n, once it has occurred for some m that p(m,n+1) > p(m,n), then it holds for all larger m. However, I did not see a simple proof of this, nor could I easily find one on the net. Presuming it is true, then the m at which p(m,n+1) first overtakes p(m,n) is of intrinsic interest.

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

a(1) = 4 since p(4,2) = 2, which is greater than p(4,1) = 1, whereas for any lesser integer, e.g. 3, p(3,2) <= p(3,1).

MATHEMATICA

t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, n - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0]; Table[m = 1; While[t[m, n + 1] <= t[m, n], m++]; m, {n, 0, 50}] (* Michael De Vlieger, Jun 23 2016, after Mats Granvik at A008284 *)

PROG

(Python)

element = 1

goal = 64

n = 1

p = [[]]

while element <= goal:

    # fill in the nth row of the table

    p.append([0]*(goal+2))

    for k in xrange(1, min(n, goal+1)+1):

        if (k == 1) or (k == n):

            p[n][k] = 1

        else:

            p[n][k] = p[n-1][k-1] + p[n-k][k]

      # see if we can increment element

    if p[n][element+1] > p[n][element]:

        print "p[{}][{}]={} and p[{}][{}]={} so a[{}] = {}".format(

            n, element, p[n][element], n, element+1, p[n][element+1], element, n)

        element = element+1

    n = n+1

CROSSREFS

Cf. A008284.

Sequence in context: A064368 A026372 A014690 * A310704 A126891 A310705

Adjacent sequences:  A274380 A274381 A274382 * A274384 A274385 A274386

KEYWORD

nonn

AUTHOR

Glen Whitney, Jun 23 2016

STATUS

approved

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Last modified May 19 17:48 EDT 2019. Contains 323395 sequences. (Running on oeis4.)