A180291 -id:A180291 - cp's OEIS frontend

A060775 The greatest divisor d|n such that d < n/d, with a(1) = 1.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 1, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 4, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 3, 2, 1, 7, 5, 2, 3

Offset: 1

Labos Elemer, Apr 26 2001

Also: Largest divisor of n which is less than sqrt(n).
If n is not a square, then a(n) = A033676(n), else a(n) is strictly smaller than A033676(n) = sqrt(n) (except for a(1) = 1).  - M. F. Hasler, Sep 20 2011
Record values occur for n = k * (k+1), for which a(n) = k. - Franklin T. Adams-Watters, May 01 2015
If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence gives the greatest strictly inferior divisor, which may differ from the lower central divisor A033676. Central divisors are listed by A207375. - Gus Wiseman, Feb 28 2021

			n = 252, D = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, 18 divisors, the 9th is 14, so a(252) = 14.
From _Gus Wiseman_, Feb 28 2021: (Start)
The strictly inferior divisors of selected n:
n = 1  2  6  12  20  30  42  56  72  90  110  132  156  182  210  240
    -----------------------------------------------------------------
    {} 1  1  1   1   1   1   1   1   1   1    1    1    1    1    1
          2  2   2   2   2   2   2   2   2    2    2    2    2    2
             3   4   3   3   4   3   3   5    3    3    7    3    3
                     5   6   7   4   5   10   4    4    13   5    4
                                 6   6        6    6         6    5
                                 8   9        11   12        7    6
                                                             10   8
                                                             14   10
                                                                  12
                                                                  15
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 2..1000 from Harry J. Smith)

Cf. A033677, A000196, A000005, A000142, A027423, A055226, A060776, A060777, A002378.
The weakly inferior version is A033676.
Positions of first appearances are A180291.
These are the row-maxima of A341674.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A070039 adds up strictly inferior divisors.
A207375 lists central divisors.
A333805 counts strictly inferior odd divisors.
A333806 counts strictly inferior prime divisors.
A341596 counts strictly inferior squarefree divisors.
A341677 counts strictly inferior prime-power divisors.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A161908, A341591.
- Strictly Superior: A048098, A064052, A140271, A238535, A341642, A341673.
Cf. A000203, A001248, A006530, A020639, A112798, A161901.

with(numtheory):
a:= n-> max(select(d-> is(d=1 or dAlois P. Heinz, Jan 29 2018

Table[Part[Divisors[w], Floor[DivisorSigma[0, w]/2]], {w, 1, 256}]
Table[If[n==1,1,Max[Select[Divisors[n],#Gus Wiseman, Feb 28 2021 *)

A060775(n)=if(n>1,divisors(n)[numdiv(n)\2],1) \\ M. F. Hasler, Sep 21 2011

a(n) = max { d: d|n and d < sqrt(n) or d = 1 }, where "|" means "divides". [Corrected by M. F. Hasler, Apr 03 2019]

a(1) = 1 added (to preserve the relation a(n) | n) by Franklin T. Adams-Watters, Jan 27 2018
Edited by M. F. Hasler, Apr 03 2019
Name changed by Gus Wiseman, Feb 28 2021 (was: Lower central (median) divisor of n, with a(1) = 1.)

A180281 Triangle read by rows: T(n,k) = number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to k.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 18, 12, 4, 1, 50, 50, 20, 5, 1, 140, 195, 90, 30, 6, 1, 392, 735, 392, 147, 42, 7, 1, 1106, 2716, 1652, 672, 224, 56, 8, 1, 3138, 9912, 6804, 2970, 1080, 324, 72, 9, 1, 8952, 35850, 27600, 12825, 4950, 1650, 450, 90, 10, 1, 25652, 128865, 110715, 54450, 22022, 7865, 2420, 605, 110, 11

Offset: 1

R. H. Hardin, Aug 24 2010

To clarify a slight ambiguity in the definition, the heaviest box in such an arrangement should contain exactly k balls. - Gus Wiseman, Sep 22 2016

			The T(4,2)=18 arrangements are {0022, 0112, 0121, 0202, 0211, 0220, 1012, 1021, 1102, 1120, 1201, 1210, 2002, 2011, 2020, 2101, 2110, 2200}.
Triangle starts
  1
  1   2
  1   6   3
  1  18  12  4
  1  50  50 20  5
  1 140 195 90 30 6
  ...

Alois P. Heinz, Rows n = 1..200, flattened (first 59 rows from R. H. Hardin)

Cf. A097861, A180282, A180291.
Row sums give A088218.
T(n,ceiling(n/2)) gives A318160.
T(2n,n) gives A318161.
T(2n-1,n) gives A318161.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, 0, add(b(n-j, i-1, k), j=0..min(n, k))))
    end:
T:= (n, k)-> b(n$2, k)-b(n$2, k-1):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 16 2018
# second Maple program:
T:= (n, k)-> coeff(series(((x^(k+1)-1)/(x-1))^n
             -((x^k-1)/(x-1))^n, x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 17 2018

T[n_,k_]:=Select[Tuples[Range[0,k],n],And[Max[#]===k,Total[#]===n]&]; (* Gus Wiseman, Sep 22 2016 *)
SequenceForm@@@T[4,2] (* example *)
Join@@Table[Length[T[n,k]],{n,1,6},{k,1,n}] (* sequence *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, 0, Sum[b[n-j, i-1, k], {j, 0, Min[n, k]}]]];
T[n_, k_] := b[n, n, k] - b[n, n, k-1];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Aug 28 2022, after Alois P. Heinz *)

Empirical: right half of table, T(n,k) = n*binomial(2*n-k-2,n-2) for 2*k > n; also, T(n,2) = Sum_{j=1..n} binomial(n,j)*binomial(n-j,j) = 2*A097861(n). - Robert Gerbicz in the Sequence Fans Mailing List
From Alois P. Heinz, Aug 17 2018: (Start)
T(n,k) = [x^n] ((x^(k+1)-1)/(x-1))^n - ((x^k-1)/(x-1))^n.
T(n,k) = A305161(n,k) - A305161(n,k-1). (End)

A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 2, 12, 12, 3, 108, 120, 24, 4, 1280, 1520, 280, 40, 5, 18750, 23400, 3930, 510, 60, 6, 326592, 423360, 65016, 7644, 840, 84, 7, 6588344, 8800008, 1241464, 132552, 13440, 1288, 112, 8, 150994944, 206622720, 26911296, 2622528, 244944, 22032, 1872, 144, 9

Offset: 1

Walt Rorie-Baety, Aug 15 2013

			T(1,1) =  1: [1].
T(2,1) =  2: [1,2], [2,1].
T(2,2) =  2: [1,1], [2,2].
T(3,1) = 12: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3].
T(3,2) = 12: [1,1,2], [1,1,3], [1,2,2], [1,3,3], [2,1,1], [2,2,1], [2,2,3], [2,3,3], [3,1,1], [3,2,2], [3,3,1], [3,3,2].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
.       1;
.       2,       2;
.      12,      12,       3;
.     108,     120,      24,      4;
.    1280,    1520,     280,     40,     5;
.   18750,   23400,    3930,    510,    60,    6;
.  326592,  423360,   65016,   7644,   840,   84,   7;
. 6588344, 8800008, 1241464, 132552, 13440, 1288, 112,  8;

Alois P. Heinz, Rows n = 1..141, flattened
Project Euler, Problem 427: n-sequences

Row sums give: A000312.
Column k=1 gives: A055897.
Main diagonal gives: A000027.
Lower diagonal gives: 2*A180291.
Cf. A228194, A228273, A228617.

T:= proc(n) option remember; local b; b:=
      proc(m, s, i) option remember; `if`(m>i or s>m, 0,
        `if`(i=1, n, `if`(s=1, (n-1)*add(b(m, h, i-1), h=1..m),
         b(m, s-1, i-1) +`if`(s=m, b(m-1, s-1, i-1), 0))))
      end; forget(b);
      seq(add(b(k, s, n), s=1..k), k=1..n)
    end:
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 18 2013

T[n_] := T[n] = Module[{b}, b[m_, s_, i_] := b[m, s, i] = If[m>i || s>m, 0, If[i == 1, n, If[s == 1, (n-1)*Sum[b[m, h, i-1], {h, 1, m}], b[m, s-1, i-1] + If[s == m, b[m-1, s-1, i-1], 0]]]]; Table[Sum[b[k, s, n], {s, 1, k}], {k, 1, n}]]; Table[ T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A228194(n). - Alois P. Heinz, Dec 23 2020

A174345 Triangle T(n, k) = (1/k)binomial(n-1, k-1)binomial(n, k-1)2^(k-1) if floor(n/2) >= k, otherwise (1/k)binomial(n-1, k-1)binomial(n, k-1)2^(n-k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 12, 12, 1, 1, 20, 80, 20, 1, 1, 30, 200, 200, 30, 1, 1, 42, 420, 1400, 420, 42, 1, 1, 56, 784, 3920, 3920, 784, 56, 1, 1, 72, 1344, 9408, 28224, 9408, 1344, 72, 1, 1, 90, 2160, 20160, 84672, 84672, 20160, 2160, 90, 1

Offset: 1

Roger L. Bagula, Mar 16 2010

			Triangle begins as:
  1;
  1,  1;
  1,  6,    1;
  1, 12,   12,     1;
  1, 20,   80,    20,     1;
  1, 30,  200,   200,    30,     1;
  1, 42,  420,  1400,   420,    42,     1;
  1, 56,  784,  3920,  3920,   784,    56,    1;
  1, 72, 1344,  9408, 28224,  9408,  1344,   72,  1;
  1, 90, 2160, 20160, 84672, 84672, 20160, 2160, 90,  1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000217, A180291.

Table[(Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>=k, 2^(k-1), 2^(n-k)], {n,12}, {k,n}]//Flatten

def A174345(n,k):
    b=binomial
    if ((n//2)>k-1): return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(k-1)
    else: return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(n-k)
flatten([[A174345(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Nov 28 2021

T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k).
T(n, n-k) = T(n, k).
From G. C. Greubel, Nov 28 2021: (Start)
T(n, n-1) = A180291(n), n > 1.
T(n, n-1) = 2*A000217(n-1), n > 2. (End)

Edited by G. C. Greubel, Nov 28 2021

cp's OEIS Frontend

A060775 The greatest divisor d|n such that d < n/d, with a(1) = 1.

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A180281 Triangle read by rows: T(n,k) = number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to k.

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A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A174345 Triangle T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k), read by rows.

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A174345 Triangle T(n, k) = (1/k)binomial(n-1, k-1)binomial(n, k-1)2^(k-1) if floor(n/2) >= k, otherwise (1/k)binomial(n-1, k-1)binomial(n, k-1)2^(n-k), read by rows.